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File: Solving Inequalities Pdf 176004 | 7 4 Solving Logarithmic Equations And Inequalities
solve each equation 1 solution 2 solution 7 4 solving logarithmic equations and inequalities 2 solve each equation multiple choice 3 solve log5 x 10 log5 3x 1 a 10 ...

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                                                                                                                 Solve each equation.
                                                                                                                                  
                                                                                                              1. 
                                                                                                                 SOLUTION:  
                                                                                                                                        
                                                                                                                                   
                                                                                                              2. 
                                                                                                                 SOLUTION:  
                                                                                                                                        
            7-4 Solving Logarithmic Equations and Inequalities
                                                                                                 
                                                                                                                                                                          2
                   Solve each equation.                                                                                                                                                        
                                                                                                                 MULTIPLE CHOICE
                                                                                                              3.                                       Solve log5 (x − 10) = log5
                                                                                                                       
                                                                                                                 3x.
                1. 
                                                                                                                 A 10 
                                                                                                                 B   
                   SOLUTION:                                                                                         2
                                                                                                                 C   
                                                                                                                     5
                                                                                                                 D 2, 5 
                                                                                                                 SOLUTION:  
                                                                                                                                                     
                                                                                                                   
                                                                                                                                                       
                2. 
                   SOLUTION:                                                                                       
                                                                                                                 Substitute each value into the original equation. 
                                                                                                                   
                                                                                                                                                                              
                                                                                                                   
                                                                                                                 The domain of a logarithmic function cannot be 0, so 
                                                                                                                 log5 (–6) is undefined and –2 is an extraneous 
                                                                            2                                    solution. 
                   MULTIPLE CHOICE
                3.                                       Solve log5 (x − 10) = log5                                
                         
                   3x.                                                                                           C is the correct option. 
                   A 10 
                   B                                                                                             Solve each inequality.
                       2                                                                                                      
                   C                                                                                          4. log    x > 3
                       5                                                                                              5
                   D 2, 5 
                                                                                                                 SOLUTION:  
                   SOLUTION:                                                                                                        
                                                       
                                                                                                                   
                                                                                                                                                          
                                                                                                                 Thus, solution set is  {x | x > 125}.
                                                                                                                                   
                                                                                                              5. log8 x ≤ −2
                                                                                                                 SOLUTION:  
                                                                                                                                   
                   Substitute each value into the original equation. 
                     
                                                                                
            eSolutions Manual - Powered by Cognero                                                                                                                                      Page1
                                                                                                                 Thus, solution set is                                 .
                   The domain of a logarithmic function cannot be 0, so 
                   log5 (–6) is undefined and –2 is an extraneous                                                                                          
                   solution.                                                                                  6. log4 (2x + 5) ≤ log4 (4x − 3)
                                                                                                                 SOLUTION:  
                   C is the correct option.                                                                                                                 
                   Solve each inequality.
                                
                4. log5 x > 3
                   SOLUTION:                                                                                       
                                                                                                                                                         
                                                                                                                 Thus, solution set is {x | x ≥ 4}.
                                                                                                                                                      
                                                                                                              7. log8 (2x) > log8 (6x − 8)
                     
                                                                                                                 SOLUTION:  
                   Thus, solution set is  {x | x > 125}.                                                                                              
                                     
                5. log8 x ≤ −2
                   SOLUTION:  
                                                                                                                   
                                                                                                                 Exclude all values of x for which 
                                                                                                                                                 
                                                                                                                   
                                                                                                                 So,                                     
                   Thus, solution set is                                 .                                         
                                                                                                                 Thus, solution set is                               . 
                                                             
                6. log4 (2x + 5) ≤ log4 (4x − 3)
                                                                                                                 CCSS STRUCTURE  Solve each equation.
                   SOLUTION:                                                                                                       
                                                              
                                                                                                              8. 
                                                                                                                 SOLUTION:  
                                                                                                                                        
                     
                                                           
                   Thus, solution set is {x | x ≥ 4}.
                                                        
                7. log8 (2x) > log8 (6x − 8)
                   SOLUTION:  
                                                        
                                                                                                                                   
                                                                                                              9. 
                     
                                                                                                                 SOLUTION:  
                   Exclude all values of x for which                                                                                     
                                                   
                     
                   So,                                     
                     
                   Thus, solution set is                               . 
                   CCSS STRUCTURE  Solve each equation.                                                                           
                                                                                                            10. 
                8. 
                                                                                                                 SOLUTION:  
                   SOLUTION:                                                                                                         
                                          
                                                                                                                                    
                                                                                                            11. 
                9. 
                                                                                                                 SOLUTION:  
                   SOLUTION:                                                                                                           
                                           
                                                                                                                                    
                                                                                                            12. 
                                    
                                                                                                                 SOLUTION:  
              10.                                                                                                                    
                   SOLUTION:  
                                       
                                                                                                                                    
                                                                                                            13. 
                                      
              11. 
                                                                                                                 SOLUTION:  
                                                                                                                                     
                   SOLUTION:  
                                         
                                      
              12.                                                                                                                                 2        
                                                                                                            14. log3 (3x + 8) = log3 (x + x)
                   SOLUTION:  
                                                                                                                 SOLUTION:  
                                                                                                                                                              
                                                                                                                   
                                                                                                                                                   
                                                                                                                   
                                                                                                                 Substitute each value into the original equation.
              13.                                                                                                  
                                                                                                                                                                              
                   SOLUTION:  
                                                                                                                   
                                                                                                                 Thus, x = –2 or 4. 
                                                                                                                             2                              
                                                                                                            15. log12 (x − 7) = log12 (x + 5)
                                                                                                                 SOLUTION:  
                                                                                                                                                            
                                                    2        
              14. log3 (3x + 8) = log3 (x + x)
                                                                                                                   
                                                                                                                  
                   SOLUTION:                                                                                                                       
                                                                
                                                                                                                   
                                                                                                                 Substitute each value into the original equation.
                                                                                                                   
                                                                                                                                                                              
                     
                                                     
                                                                                                                   
                                                                                                                 Thus, x = –3 or 4. 
                   Substitute each value into the original equation.                                                       2                           
                                                                                                            16. log6 (x − 6x) = log6 (−8)
                                                                                
                                                                                                                 SOLUTION:  
                                                                                                                                                         
                     
                   Thus, x = –2 or 4. 
                               2                              
              15. log12 (x − 7) = log12 (x + 5)
                                                                                                                   
                                                                                                                  
                   SOLUTION:                                                                                                                     
                                                              
                                                                                                                   
                                                                                                                 Substitute each value into the original equation. 
                                                                                                                   
                                                                                                                                                                              
                    
                                                     
                                                                                                                   
                   Substitute each value into the original equation.                                             log6 (–8) is undefined, so 4 and 2 are extraneous 
                                                                                                                 solutions. 
                                                                                                                 Thus, no solution. 
                                                                                                                           2                                   
                                                                                                            17. log9 (x − 4x) = log9 (3x − 10)
                     
                   Thus, x = –3 or 4.                                                                            SOLUTION:  
                                                                                                                                                               
                             2                           
              16. log6 (x − 6x) = log6 (−8)
                   SOLUTION:  
                                                           
                                                                                                                   
                                                                                                                  
                                                                                                                                                 
                                                                                                                   
                                                                                                                 Substitute each value into the original equation. 
                    
                                                                                                                   
                                                                                                                                                                              
                     
                   Substitute each value into the original equation. 
                                                                                                                   
                                                                                                                 log9 (–4) is undefined and 2 is extraneous solution.
                                                                                                                 Thus, x = 5. 
                                                                                                                             2                                 
                                                                                                            18. log4 (2x + 1) = log4 (10x − 7)
                   log6 (–8) is undefined, so 4 and 2 are extraneous 
                   solutions.                                                                                    SOLUTION:  
                   Thus, no solution.                                                                                                                          
                             2                                   
              17. log9 (x − 4x) = log9 (3x − 10)
                   SOLUTION:  
                                                                                                                   
                                                                                                                  
                                                                                                                                                     
                                                                                                                   
                                                                                                                 Substitute each value into the original equation.
                                                                                                                   
                    
                                                                                                                                                                              
                                                                                                                   
                   Substitute each value into the original equation.                                             Thus, x = 1 or 4. 
                     
                                                                                                                           2                                
                                                                                                            19. log7 (x − 4) = log7 (− x + 2)
                                                                                                                 SOLUTION:  
                   log  (–4) is undefined and 2 is extraneous solution.                                                                                       
                        9
                   Thus, x = 5. 
                               2                                 
              18. log4 (2x + 1) = log4 (10x − 7)
                                                                                                                   
                                                                                                                  
                   SOLUTION:                                                                                                                         
                                                                 
                                                                                                                   
                                                                                                                 Substitute each value into the original equation. 
                                                                                                                   
                                                                                                                                                                              
                    
                                                       
                                                                                                                   
                                                                                                                 Since you can not have a log of 0, x =                      3 is the 
                   Substitute each value into the original equation.                                             solution. 
                                                                                                                                   
                                                                                                                 SCIENCEThe equation for wind speed w, in 
                                                                                                                 miles per hour, near the center of a tornado is w
                                                                                                                 = 93 log  d + 65, where d is the distance in 
                                                                                                                               10
                                                                                                                                                                   
                   Thus, x = 1 or 4.                                                                             miles that the tornado travels.
                                                                                                            20. Write this equation in exponential form.
                             2
                                                                                                                 SOLUTION:  
              19. log7 (x − 4) = log7 (− x + 2)
                   SOLUTION:  
                                                                
                     
                                                                                                            21. In May of 1999, a tornado devastated Oklahoma City
                                                                                                                 with the fastest wind speed ever recorded. If the 
                                                                                                                 tornado traveled 525 miles, estimate the wind speed 
                                                                                                                 near the center of the tornado. 
                   Substitute each value into the original equation. 
                                                                                                                 SOLUTION:  
                                                                                                                 Substitute 525 for d in the equation and simplify.
                                                                                                                   
                                                                                                                                                 
                     
                   Since you can not have a log of 0, x =                      3 is the                          Solve each inequality.
                   solution.                                                                                                       
                                                                                                            22. log6 x < −3
                                     
                   SCIENCEThe equation for wind speed w, in 
                   miles per hour, near the center of a tornado is w                                             SOLUTION:  
                   = 93 log  d + 65, where d is the distance in                                                                      
                                 10
                                                                     
                   miles that the tornado travels.
              20. Write this equation in exponential form.
                   SOLUTION:  
                                                                                                                 The solution set is                                  .
                                                                                                                              
                                                                                                            23. log4 x ≥ 4
                                                                                                                 SOLUTION:  
                                                                                                                                     
              21. In May of 1999, a tornado devastated Oklahoma City
                   with the fastest wind speed ever recorded. If the 
                   tornado traveled 525 miles, estimate the wind speed                                             
                   near the center of the tornado.                                                                                                     
                                                                                                                 The solution set is {x | x ≥ 256}.
                   SOLUTION:                                                                                                       
                   Substitute 525 for d in the equation and simplify.                                       24. log3 x ≥ −4
                     
                                                                                                                 SOLUTION:  
                                                                                                                                    
                   Solve each inequality.
                                     
              22. log6 x < −3
                                                                                                                 The solution set is                          .
                   SOLUTION:  
                                       
                                                                                                                                   
                                                                                                            25. log2 x ≤ −2
                                                                                                                 SOLUTION:  
                                                                                                                                    
                   The solution set is                                  .
                                
              23. log4 x ≥ 4
                                                                                                                 The solution set is                              .
                   SOLUTION:  
                                                                                                                              
                                                                                                            26. log5 x > 2
                                                                                                                 SOLUTION:  
                                                                                                                                   
                                                         
                   The solution set is {x | x ≥ 256}.
                                     
              24. log3 x ≥ −4                                                                                      
                   SOLUTION:                                                                                     The solution set is                         .
                                      
                                                                                                                                   
                                                                                                            27. log7 x < −1
                                                                                                                 SOLUTION:  
                                                                                                                                   
                   The solution set is                          .
                                     
              25. log2 x ≤ −2
                                                                                                                 The solution set is                              .
                   SOLUTION:  
                                      
                                                                                                                                                           
                                                                                                            28. log2 (4x − 6) > log2 (2x + 8)
                                                                                                                 SOLUTION:  
                                                                                                                                                            
                   The solution set is                              .
                                
              26. log5 x > 2
                                                                                                                   
                   SOLUTION:                                                                                     The solution set is                        .
                                     
                                                                                                                                                           
                                                                                                            29. log7 (x + 2) ≥  log7 (6x − 3)
                                                                                                                 SOLUTION:  
                   The solution set is                         .                                                                                          
                                     
              27. log7 x < −1
                   SOLUTION:                                                                                       
                                                                                                                 Exclude all values of x for which 
                                                                                                                                                    
                                                                                                                   
                                                                                                                 So,                                       
                   The solution set is                              .                                              
                                                                                                                 The solution set is                             . 
                                                             
              28. log2 (4x − 6) > log2 (2x + 8)
                                                                                                                                                           
                   SOLUTION:                                                                                30. log3 (7x – 6) < log3 (4x + 9)
                                                              
                                                                                                                 SOLUTION:  
                                                                                                                                                            
                     
                   The solution set is                        .                                                    
                                                                                                                 Exclude all values of x for which 
              29. log7 (x + 2) ≥  log7 (6x − 3)                                                                                                       
                                                                                                                   
                   SOLUTION:  
                                                                                                                 So,                                        
                                                                                                                   
                                                                                                                 The solution set is                              . 
                     
                   Exclude all values of x for which                                                                                                         
                                                                                                            31. log5 (12x + 5) ≤ log5 (8x + 9)
                     
                                                                                                                 SOLUTION:  
                   So,                                                                                                                                       
                     
                   The solution set is                             . 
                                                                                                                   
                                                                                                                 Exclude all values of x for which 
              30. log3 (7x – 6) < log3 (4x + 9)                                                                                                        
                                                                                                                   
                   SOLUTION:                                                                                                                                    
                                                                                                                 So, 
                                                                                                                   
                                                                                                                 The solution set is                                  . 
                     
                   Exclude all values of x for which                                                                                                          
                                                                                                            32. log11 (3x − 24) ≥ log11 (−5x − 8)
                     
                                                                                                                 SOLUTION:  
                   So, 
                     
                   The solution set is                              . 
                                                                                                                   
                                                                                                                                                              
                                                                                                                 The solution set is {x | x ≥ 2}.
              31. log5 (12x + 5) ≤ log5 (8x + 9)
                                                                                                                                                         
                                                                                                            33. log     (9x + 4) ≤ log (11x − 12)
                   SOLUTION:                                                                                          9                     9
                                                               
                                                                                                                 SOLUTION:  
                     
                   Exclude all values of x for which                                                               
                                                                                                                                                       
                                                                                                                 The solution set is {x | x ≥ 8}.
                                                                                                                                                  
                   So,                                                                                      34. CCSS MODELINGThe magnitude of an 
                                                                                                                 earthquake is measured on a logarithmic scale called 
                                                                                                                                                                                               
                                                                                                                 the Richter scale. The magnitude M is given by M =
                   The solution set is                                  .                                        log10 x, where x represents the amplitude of the 
                                                                                                                 seismic wave causing ground motion. 
                                                                                                                 a. How many times as great is the amplitude caused 
              32. log11 (3x − 24) ≥ log11 (−5x − 8)                                                              by an earthquake with a Richter scale rating of 8 as 
                                                                                                                 an aftershock with a Richter scale rating of 5? 
                   SOLUTION:                                                                                     b. In 1906, San Francisco was almost completely 
                                                                                                                 destroyed by a 7.8 magnitude earthquake. In 1911, 
                                                                                                                 an earthquake estimated at magnitude 8.1 occurred 
                                                                                                                 along the New Madrid fault in the Mississippi River 
                                                                                                                 Valley. How many times greater was the New 
                                                                                                                 Madrid earthquake than the San Francisco 
                                                                                                                 earthquake? 
                                                                
                   The solution set is {x | x ≥ 2}.
                                                                                                                 SOLUTION:  
                                                                                                                 a. 
              33. log9 (9x + 4) ≤ log9 (11x − 12)                                                                The amplitude of the seismic wave with a Richter 
                                                                                                                                                            8           5                     
                   SOLUTION:                                                                                     scale rating of 8 and 5 are 10  and 10  respectively.
                                                                                                                 Divide 108 by 105. 
                                                                                                                   
                                                                                                                                   
                                                                                                                   
                                                         
                   The solution set is {x | x ≥ 8}.                                                                                                    3
                                                                                                                 The scale rating of 8 is 10  or 1000 times greater 
              34. CCSS MODELINGThe magnitude of an                                                               than the scale rating of 5. 
                   earthquake is measured on a logarithmic scale called                                            
                                                                                                 
                   the Richter scale. The magnitude M is given by M =                                            b. 
                   log10 x, where x represents the amplitude of the                                              The amplitudes of San Francisco earthquake and 
                   seismic wave causing ground motion.                                                           New Madrid earthquake were 107.8 and 108.1 
                   a. How many times as great is the amplitude caused                                            respectively. 
                   by an earthquake with a Richter scale rating of 8 as                                            
                   an aftershock with a Richter scale rating of 5?                                               Divide 108.1 by 107.8. 
                   b. In 1906, San Francisco was almost completely                                                 
                   destroyed by a 7.8 magnitude earthquake. In 1911,                                              
                   an earthquake estimated at magnitude 8.1 occurred                                                                    
                   along the New Madrid fault in the Mississippi River 
                   Valley. How many times greater was the New                                                      
                   Madrid earthquake than the San Francisco                                                       
                   earthquake?                                                                                   The New Madrid earthquake was 100.3 or about 2 
                                                                                                                 times greater than the San Francisco earthquake. 
                   SOLUTION:  
                   a.                                                                                                          
                                                                                                                 MUSICThe first key on a piano keyboard 
                   The amplitude of the seismic wave with a Richter                                         35. 
                   scale rating of 8 and 5 are 108 and 105 respectively.                                         corresponds to a pitch with a frequency of 27.5 
                                 8          5                                                                    cycles per second. With every successive key, going 
                   Divide 10  by 10 .                                                                            up the black and white keys, the pitch multiplies by a 
                                                                                                                 constant. The formula for the frequency of the pitch 
                                                                                                                 sounded when the nth note up the keyboard is played
                                                                                                                 is given by                                     
                                                                                                                 a. A note has a frequency of 220 cycles per second. 
                                                         3                                                       How many notes up the piano keyboard is this? 
                   The scale rating of 8 is 10  or 1000 times greater                                            b. Another pitch on the keyboard has a frequency of 
                   than the scale rating of 5.                                                                   880 cycles per second. After how many notes up the
                                                                                                                 keyboard will this be found? 
                   b. 
                   The amplitudes of San Francisco earthquake and                                                SOLUTION:  
                   New Madrid earthquake were 107.8 and 108.1                                                    a. 
                   respectively.                                                                                 Substitute 220 for f  in the formula and solve for n.
                                                                                                                   
                                                                                                                  
                   Divide 108.1 by 107.8.                                                                                                       
                     
                    
                                          
                                                                                                                 b. 
                                                                                                                 Substitute 880 for f  in the formula and solve for n.
                   The New Madrid earthquake was 100.3 or about 2 
                                                                                                                   
                   times greater than the San Francisco earthquake.                                               
                                                                                                                                                
                                 
                   MUSICThe first key on a piano keyboard 
              35. 
                   corresponds to a pitch with a frequency of 27.5 
                   cycles per second. With every successive key, going 
                   up the black and white keys, the pitch multiplies by a 
                   constant. The formula for the frequency of the pitch 
                   sounded when the nth note up the keyboard is played                                                                                                       
                                                                                                                 MULTIPLE REPRESENTATIONSIn this 
                   is given by                                                                              36.                                                                   
                                                                                                                 problem, you will explore the graphs shown:
                                                                                                                                                         
                   a. A note has a frequency of 220 cycles per second.                                           y = log4 x and 
                   How many notes up the piano keyboard is this?                                                   
                   b. Another pitch on the keyboard has a frequency of                                            
                   880 cycles per second. After how many notes up the
                   keyboard will this be found? 
                   SOLUTION:  
                   a. 
                   Substitute 220 for f  in the formula and solve for n.
                     
                                                                                                                                                      
                                                                                                                   
                                                                                                                  
                                                                                                                 a. ANALYTICAL
                                                                                                                                               How do the shapes of the 
                                                                                                                 graphs compare? How do the asymptotes and the x-
                                                                                                                 intercepts of the graphs compare? 
                                                                                                                 b. VERBAL
                                                                                                                                      Describe the relationship between the 
                                                                                                                 graphs.
                   b.                                                                                                
                                                                                                                 c. GRAPHICAL
                   Substitute 880 for f  in the formula and solve for n.                                                                     Use what you know about 
                                                                                                                 transformations of graphs to compare and contrast 
                                                                                                                                                                                              
                                                                                                                 the graph of each function and the graph of y = log4
                                                                                                                 x. 
                                                                                                                   
                                                                                                                  
                                                                                                                                        
                                                                                                                 1. y = log4 x + 2
                                                                                                                                             
                                                                                                                 2. y = log4 (x + 2)
                                                                                                                                      
                                                                                                                 3. y = 3 log4 x
                                                                                                                   
                                                                                                                  
                   MULTIPLE REPRESENTATIONSIn this 
              36.                                                                                                d. ANALYTICAL
                   problem, you will explore the graphs shown:                                                                                  Describe the relationship 
                                                                                                                                                                      
                                                                                                                 between y = log x and y =  1(log x). What are a 
                   y = log4 x and                                                                                                        4                −        4
                                                                                                                 reasonable domain and range for each function? 
                                                                                                                     
                                                                                                                 e.ANALYTICAL
                                                                                                                                                Write an equation for a function 
                                                                                                                 for which the graph is the graph of y = log3 x 
                                                                                                                 translated 4 units left and 1 unit up. 
                                                                                                                 SOLUTION:  
                                                                                                                 a. 
                                                                                                                 The shapes of the graphs are the same. The 
                                                                                                                 asymptote for each graph is the y-axis and the x-
                                                                                                                                                             
                                                                                                                 intercept for each graph is 1.
                   a. ANALYTICAL                                                                                 b. 
                                                 How do the shapes of the                                        The graphs are reflections of each other over the x-
                   graphs compare? How do the asymptotes and the x-                                              axis. 
                   intercepts of the graphs compare?                                                                 
                   b. VERBAL                                                                                     c.
                                        Describe the relationship between the                                    1. The second graph is the same as the first, except it
                   graphs.                                                                                       is shifted horizontally to the left 2 units. 
                       
                   c. GRAPHICAL
                                               Use what you know about                                             
                   transformations of graphs to compare and contrast                                              
                                                                                                
                   the graph of each function and the graph of y = log4
                   x. 
                     
                    
                                          
                   1. y = log4 x + 2
                                               
                   2. y = log4 (x + 2)
                                        
                   3. y = 3 log x                                                                                                                                         
                                    4                                                                                  
                                                                                                                   
                                                                                                                  
                                                                                                                                                                    
                                                                                                                             2, 8] scl: 1 by [ 5, 5] scl: 1
                   d. ANALYTICAL                                                                                         [−                       −
                                                  Describe the relationship                                        
                                                                                                                  
                   between y = log x and y =  1(log x). What are a 
                                           4                −        4                                           2. The second graph is the same as the first, except it
                   reasonable domain and range for each function?                                                is shifted vertically up 2 units. 
                       
                   e.ANALYTICAL
                                                  Write an equation for a function                                 
                   for which the graph is the graph of y = log  x                                                 
                                                                                 3
                   translated 4 units left and 1 unit up. 
                   SOLUTION:  
                   a. 
                   The shapes of the graphs are the same. The 
                   asymptote for each graph is the y-axis and the x-
                   intercept for each graph is 1.                                                                                                                       
                   b.                                                                                                  
                                                                                                                   
                   The graphs are reflections of each other over the x-                                           
                                                                                                                            4, 8] scl: 1 by [ 5, 5] scl: 1 
                   axis.                                                                                                [−                       −
                                                                                                                   
                                                                                                                  
                   c.                                                                                            3. Each point on the second graph has a y-coordinate
                   1. The second graph is the same as the first, except it                                       3 times that of the corresponding point on the first 
                   is shifted horizontally to the left 2 units.                                                  graph. 
                     
                                                                                                                   
                                                                                                                  
                                                                                                                                                                       
                                                                                                                       
                     
                                                                                                                   
                                                                                                                  
                               2, 8] scl: 1 by [ 5, 5] scl: 1                                                                                                     
                           [−                       −                                                                      2, 8] scl: 1 by [ 5, 5] scl: 1
                                                                                                                       [−                       −
                                                                                                                     
                                                                                                                 d.
                   2. The second graph is the same as the first, except it                                       The graphs are reflections of each other over the x-
                   is shifted vertically up 2 units.                                                             axis. 
                                                                                                                                                                            
                                                                                                                 D={x|x>0}; R={all real numbers}
                    
                                                                                                                 e.                                   where h is the horizontal 
                                                                                                                 shift and k is the vertical shift.  Since there is a 
                                                                                                                 horizontal shift of 4 and vertical shift of 1, h = 4 and 
                                                                                                                 k = 1. 
                                                                                                                 y = log3 (x + 4) + 1 
                                                                                                                                
                                                                                                                 SOUNDThe relationship between the intensity of 
                                                                                                            37. 
                                                                                                                 sound I and the number of decibels   
                                                                                                                                                                      β
                    
                              4, 8] scl: 1 by [ 5, 5] scl: 1 
                          [−                       −                                                             is                               , where I is the intensity of 
                     
                                                                                                                                                                  
                   3. Each point on the second graph has a y-coordinate                                          sound in watts per square meter.
                                                                                                                   
                   3 times that of the corresponding point on the first                                           
                   graph.                                                                                        a. Find the number of decibels of a sound with an 
                     
                                                                                                                 intensity of 1 watt per square meter. 
                                                                                                                 b. Find the number of decibels of sound with an 
                                                                                                                                      −2                                      
                                                                                                                 intensity of 10          watts per square meter.
                                                                                                                     
                                                                                                                 c. The intensity of the sound of 1 watt per square 
                                                                                                                 meter is 100 times as much as the intensity of 10−2 
                                                                                                                 watts per square meter. Why are the decibels of 
                                                                                                                 sound not 100 times as great? 
                         
                                                                                                                 SOLUTION:  
                    
                             2, 8] scl: 1 by [ 5, 5] scl: 1                                                      a.
                         [−                       −                                                                  
                   d.                                                                                            Substitute 1 for I in the given equation and solve for 
                   The graphs are reflections of each other over the x-                                            . 
                   axis.                                                                                         β
                                                                                                                   
                                                                                                                  
                   D={x|x>0}; R={all real numbers}                                                                                              
                   e.                                   where h is the horizontal 
                   shift and k is the vertical shift.  Since there is a 
                   horizontal shift of 4 and vertical shift of 1, h = 4 and 
                   k = 1. 
                   y = log3 (x + 4) + 1 
                                  
                   SOUNDThe relationship between the intensity of 
              37.                                                                                                  
                                                                                                                  
                   sound I and the number of decibels β                                                          b. 
                   is                               , where I is the intensity of                                Substitute 10−2 for I in the given equation and solve 
                                                                                                                 for  . 
                                                                                                                       β
                   sound in watts per square meter.                                                                
                                                                                                                  
                                                                                                                                                
                   a. Find the number of decibels of a sound with an 
                   intensity of 1 watt per square meter. 
                   b. Find the number of decibels of sound with an 
                                        −2                                      
                   intensity of 10          watts per square meter.
                       
                   c. The intensity of the sound of 1 watt per square 
                   meter is 100 times as much as the intensity of 10−2 
                                                                                                                   
                   watts per square meter. Why are the decibels of                                                
                   sound not 100 times as great?                                                                     
                                                                                                                 c. Sample answer: The power of the logarithm only 
                                                                                                                 changes by 2. The power is the answer to the 
                   SOLUTION:                                                                                     logarithm. That 2 is multiplied by the 10 before the 
                   a.                                                                                            logarithm. So we expect the decibels to change by 
                   Substitute 1 for I in the given equation and solve for                                        20. 
                     . 
                   β
                     
                                                                                                                 CCSS CRITIQUE  
                                                                                                            38.                                  Ryan and Heather are solving 
                                                                                                                 log  x    3. Is either of them correct? Explain your 
                                                                                                                      3    ≥−
                                                                                                                 reasoning. 
                                                                                                                   
                                                                                                                  
                     
                    
                   b.                                                                                                                                               
                                      −2                                                                           
                   Substitute 10          for I in the given equation and solve                                   
                   for  . 
                         β
                     
                    
                                                  
                                                                                                                                                                    
                                                                                                                 SOLUTION:  
                                                                                                                 Sample answer: Ryan; Heather did not need to 
                                                                                                                 switch the inequality symbol when raising to a 
                                                                                                                 negative power. 
                    
                       
                   c. Sample answer: The power of the logarithm only                                                                                                                          
                                                                                                                 CHALLENGEFind log 27 +log 27 +log                                    27 +
                   changes by 2. The power is the answer to the                                             39.                                       3             9             27
                   logarithm. That 2 is multiplied by the 10 before the                                                                         
                                                                                                                 log81 27 + log243 27.
                   logarithm. So we expect the decibels to change by 
                         
                   20.                                                                                           SOLUTION:  
                   CCSS CRITIQUE  
              38.                                  Ryan and Heather are solving 
                   log  x    3. Is either of them correct? Explain your 
                        3    ≥−
                   reasoning. 
                     
                    
                                                                                                                                         
                                                                                                                 REASONINGThe Property of Inequality for 
                                                                                                            40. 
                                                                                                                                                                                              
                                                                                                                 Logarithmic Functions states that when b > 1, logb x
                                                                                                                                                       
                                                                                                                 >logby if and only if x >y. What is the case for 
                                                                                                                                                                              
                                                                                                                 when 0 < b < 1? Explain your reasoning.
                                                                                                                 SOLUTION:  
                                                                                                                                                                                        
                                                                                                                 Sample answer: When 0 < b < 1, logb x > logby if 
                                                                                                                                       
                                                                                                                 and only if x  1, logb x                                          your reasoning. All logarithmic equations are of the 
                                                         
                   >logby if and only if x >y. What is the case for                                                                       
                                                                                                                 form y = logb x.
                                                                                
                   when 0 < b < 1? Explain your reasoning.                                                       a. If the base of a logarithmic equation is greater 
                                                                                                                 than 1 and the value of x is between 0 and 1, then the
                   SOLUTION:                                                                                                                                                                   
                                                                                                                 value for y is (less than, greater than, equal to) 0.
                   Sample answer: When 0 < b < 1, logb x > logby if                                              b. If the base of a logarithmic equation is between 0 
                                         
                   and only if x 81
                                                                                                                 SOLUTION:  
                                                                                                                                   
              53. 
                   SOLUTION:  
                                                                                                                   4a + 6          a 
                                                                                                                 4        16
                   log  2401                                                                                57.             ≤
              54.       7
                                                                                                                 SOLUTION:  
                   SOLUTION:                                                                                                         
                   Solve each equation or inequality. Check your 
                                  
                   solution.
                     2x + 3            
                   5        125
              55.             ≤
                                                                                                                                      
                                                                                                            58. 
                   SOLUTION:  
                                       
                                                                                                                 SOLUTION:  
                     3x − 2           
                               
              56. 3           >81
                   SOLUTION:  
                                                                                                                                      
                                                                                                            59. 
                                                                                                                 SOLUTION:  
                     4a + 6          a 
                   4        16
              57.             ≤
                   SOLUTION:  
                                       
                                                                                                                                   
                                                                                                            60. 
                                                                                                                 SOLUTION:  
                                        
              58. 
                   SOLUTION:  
                                                                                                                 SHIPPING The height of a shipping cylinder is 4 
                                                                                                            61. 
                                                                                                                 feet more than the radius. If the volume of the 
                                                                                                                 cylinder is 5  cubic feet, how tall is it? Use the 
                                                                                                                                   π
                                                                                                                                       2     
                                                                                                                 formula V = πr h.
                                        
              59. 
                                                                                                                 SOLUTION:  
                                                                                                                 Substitute 5  for V and r + 4 for h in the formula and
                                                                                                                                  π
                   SOLUTION:                                                                                     simplify. 
                                                                                                                   
                                                                                                                  
                                                                                                                                                        
                                                                                                                   
                                                                                                                  
                                                                                                                 The equation has one real root r = 1. 
                                                                                                                 Thus, the height of the shipping cylinder is 1 + 4 = 5 
                                                                                                                      
                                                                                                                 ft.
                                                                                                                                                     
                                                                                                                 NUMBER THEORYTwo complex conjugate 
                                                                                                            62. 
              60.                                                                                                numbers have a sum of 12 and a product of 40. Find 
                                                                                                                 the two numbers 
                   SOLUTION:  
                                                                                                                 SOLUTION:  
                                                                                                                 The equations that represent the situation are:
                                                                                                                   
                                                                                                                  
                                                                                                                                                             
                                                                                                                   
                                                                                                                  
                                                                                                                 Solve equation (1). 
                                                                                                                   
                   SHIPPING The height of a shipping cylinder is 4                                                
              61.                                                                                                                                  
                   feet more than the radius. If the volume of the 
                   cylinder is 5  cubic feet, how tall is it? Use the 
                                     π
                                         2     
                   formula V = πr h.
                                                                                                                   
                                                                                                                  
                   SOLUTION:                                                                                     Solve equation (2). 
                   Substitute 5  for V and r + 4 for h in the formula and
                                    π                                                                              
                   simplify.                                                                                      
                                                                                                                                                                              
                    
                                                          
                     
                    
                   The equation has one real root r = 1.                                                           
                                                                                                                  
                   Thus, the height of the shipping cylinder is 1 + 4 = 5                                        Thus, the two numbers are 6 + 2i and 6   2i. 
                                                                                                                                                                            –
                   ft.
                                                                                                                 Simplify. Assume that no variable equals zero.
                                                                                                                   5      3
                   NUMBER THEORYTwo complex conjugate 
              62.                                                                                                x    x  
                   numbers have a sum of 12 and a product of 40. Find                                       63.       ·
                                             
                   the two numbers                                                                               SOLUTION:  
                                                                                                                                      
                   SOLUTION:  
                   The equations that represent the situation are:
                     
                    
                                                                                                                 a2   a6 
                                                                                                            64.       ·
                                                                                                                 SOLUTION:  
                                                                                                                                       
                   Solve equation (1). 
                     
                    
                                                                                                                      2    3 
                                                                                                            65. (2p n)
                                                                                                                 SOLUTION:  
                     
                    
                   Solve equation (2). 
                     
                                                                                                                 (3b3c2)2 
                                                                                                            66. 
                                                                                                                 SOLUTION:  
                                                                                                                           
                    
                   Thus, the two numbers are 6 + 2i and 6   2i.                                             67. 
                                                                              –
                   Simplify. Assume that no variable equals zero.                                                SOLUTION:  
                   x5   x3                                                                                                           
              63.       ·
                   SOLUTION:  
                                                                                                                             
                                                                                                            68. 
                     2      6
                   a    a                                                                                        SOLUTION:  
              64.       ·                                                                                                         
                   SOLUTION:  
                                         
                   (2p2n)3 
              65. 
                   SOLUTION:  
                   (3b3c2)2 
              66. 
                   SOLUTION:  
                             
              67. 
                   SOLUTION:  
                                       
                               
              68. 
                   SOLUTION:  
                                    
                                                                                                                 Solve each equation.
                                                                                                                                  
                                                                                                              1. 
                                                                                                                 SOLUTION:  
                                                                                                                                        
                                                                                                                                   
                                                                                                              2. 
                                                                                                                 SOLUTION:  
                                                                                                                                        
                                                                                                                                                                          2                    
                                                                                                                 MULTIPLE CHOICE
                                                                                                              3.                                       Solve log5 (x − 10) = log5
                                                                                                                       
                                                                                                                 3x.
                                                                                                                 A 10 
                                                                                                                 B   
                                                                                                                     2
                                                                                                                 C   
                                                                                                                     5
                                                                                                                 D 2, 5 
                                                                                                                 SOLUTION:  
                                                                                                                                                     
                   Solve each equation.                                                                            
                                                                                                                                                       
                1. 
                   SOLUTION:                                                                                       
                                                                                                                 Substitute each value into the original equation. 
                                                                                                                   
                                                                                                                                                                              
                                                                                                                   
                                                                                                                 The domain of a logarithmic function cannot be 0, so 
                                                                                                                 log5 (–6) is undefined and –2 is an extraneous 
                                                                                                                 solution. 
                                                                                                                   
                2.                                                                                               C is the correct option. 
                   SOLUTION:                                                                                     Solve each inequality.
                                                                                                                              
                                                                                                              4. log5 x > 3
                                                                                                                 SOLUTION:  
                                                                                                                                    
                                                                                                                   
                                                                                                                                                          
                                                                                                                 Thus, solution set is  {x | x > 125}.
                                                                            2                    
                   MULTIPLE CHOICE
                3.                                       Solve log5 (x − 10) = log5                                                
                                                                                                              5. log8 x ≤ −2
                   3x.
                   A 10 
                   B                                                                                             SOLUTION:  
                       2                                                                                                           
                   C   
                       5
                   D 2, 5 
                   SOLUTION:  
                                                                                                                   
                                                                                                                 Thus, solution set is                                 .
                                                                                                                                                           
                                                                                                              6. log4 (2x + 5) ≤ log4 (4x − 3)
                                                                                                                 SOLUTION:  
                                                                                                                                                            
                     
                   Substitute each value into the original equation. 
                     
                                                                                                                   
                                                                                                                                                         
                                                                                                                 Thus, solution set is {x | x ≥ 4}.
                                                                                                                                                      
                                                                                                              7. log8 (2x) > log8 (6x − 8)
                   The domain of a logarithmic function cannot be 0, so 
                   log  (–6) is undefined and –2 is an extraneous                                                SOLUTION:  
                        5                                                                                                                             
                   solution. 
                     
                   C is the correct option. 
                   Solve each inequality.                                                                          
                                
                4. log5 x > 3                                                                                    Exclude all values of x for which 
                                                                                                                                                 
                   SOLUTION:                                                                                       
                                                                                                                                                         
                                                                                                                 So, 
                                                                                                                   
            7-4 Solving Logarithmic Equations and Inequalities                                                                                                         
                                                                                                                 Thus, solution set is                               .
                   Thus, solution set is  {x | x > 125}.
                                                                                                                 CCSS STRUCTURE  Solve each equation.
                5. log8 x ≤ −2                                                                                                     
                                                                                                              8. 
                   SOLUTION:  
                                     
                                                                                                                 SOLUTION:  
                                                                                                                                        
                     
                   Thus, solution set is                                 .
                                                             
                6. log4 (2x + 5) ≤ log4 (4x − 3)
                   SOLUTION:                                                                                                       
                                                              
                                                                                                              9. 
                                                                                                                 SOLUTION:  
                                                                                                                                         
                     
                                                           
                   Thus, solution set is {x | x ≥ 4}.
                                                        
                7. log8 (2x) > log8 (6x − 8)
                   SOLUTION:  
                                                        
                                                                                                                                  
                                                                                                            10. 
                                                                                                                 SOLUTION:  
                   Exclude all values of x for which                                                                                 
                                                   
                     
                   So,                                     
                     
                   Thus, solution set is                               . 
                   CCSS STRUCTURE  Solve each equation.                                                                             
                                                                                                            11. 
                8. 
                                                                                                                 SOLUTION:  
                   SOLUTION:                                                                                                           
                                          
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                                                                                                            12. 
                                                                                                                 SOLUTION:  
                9.                                                                                                                   
                   SOLUTION:  
                                           
                                                                                                                                    
                                                                                                            13. 
                                    
              10. 
                                                                                                                 SOLUTION:  
                                                                                                                                     
                   SOLUTION:  
                                       
                                                                                                                                                  2        
                                                                                                            14. log3 (3x + 8) = log3 (x + x)
              11. 
                                                                                                                 SOLUTION:  
                                                                                                                                                              
                   SOLUTION:  
                                         
                                                                                                                   
                                                                                                                                                   
                                                                                                                   
                                                                                                                 Substitute each value into the original equation.
              12.                                                                                                  
                                                                                                                                                                              
                   SOLUTION:  
                                       
                                                                                                                   
                                                                                                                 Thus, x = –2 or 4. 
                                                                                                                             2                              
                                                                                                            15. log12 (x − 7) = log12 (x + 5)
                                                                                                                 SOLUTION:  
                                                                                                                                                            
                                      
              13. 
                                                                                                                   
                                                                                                                  
                   SOLUTION:                                                                                                                       
                                       
                                                                                                                   
                                                                                                                 Substitute each value into the original equation.
                                                                                                                   
                                                                                                                                                                              
                                                                                                                   
                                                                                                                 Thus, x = –3 or 4. 
                                                    2                                                                      2                           
              14. log3 (3x + 8) = log3 (x + x)                                                              16. log6 (x − 6x) = log6 (−8)
                   SOLUTION:  
                                                                                                                 SOLUTION:  
                                                                                                                                                         
                                                                                                                   
                                                                                                                  
                                                                                                                                                 
                                                                                                                   
                   Substitute each value into the original equation.                                             Substitute each value into the original equation. 
                                                                                                                   
                                                                                                                                                                              
                                                                                                                   
                   Thus, x = –2 or 4.                                                                                   
                                                                                                                 log6 (–8) is undefined, so 4 and 2 are extraneous 
                               2                                                                                                
              15. log12 (x − 7) = log12 (x + 5)                                                                  solutions.
                                                                                                                 Thus, no solution. 
                   SOLUTION:  
                                                                                                                           2                                   
                                                                                                            17. log9 (x − 4x) = log9 (3x − 10)
                                                                                                                 SOLUTION:  
                                                                                                                                                               
                     
                    
                                                     
                                                                                                                   
                                                                                                                  
                   Substitute each value into the original equation.                                                                             
                     
                                                                                                                   
                                                                                                                 Substitute each value into the original equation. 
                                                                                                                   
                                                                                                                                                                              
                   Thus, x = –3 or 4. 
                             2                           
              16. log6 (x − 6x) = log6 (−8)                                                                        
                                                                                                                 log9 (–4) is undefined and 2 is extraneous solution.
                   SOLUTION:                                                                                     Thus, x = 5. 
                                                           
                                                                                                                             2                                 
                                                                                                            18. log4 (2x + 1) = log4 (10x − 7)
                                                                                                                 SOLUTION:  
                                                                                                                                                               
                    
                                                   
                     
                   Substitute each value into the original equation.                                               
                                                                                                                  
                                                                                                                                                     
                                                                                                                   
                                                                                                                 Substitute each value into the original equation.
                                                                                                                   
                   log6 (–8) is undefined, so 4 and 2 are extraneous                                                                                                          
                   solutions. 
                   Thus, no solution.                                                                              
                             2                                                                                   Thus, x = 1 or 4. 
              17. log9 (x − 4x) = log9 (3x − 10)
                                                                                                                           2                                
                                                                                                            19. log     (x −4) =log (−x +2)
                   SOLUTION:                                                                                          7                     7
                                                                 
                                                                                                                 SOLUTION:  
                                                                                                                                                              
                     
                    
                                                                                                                   
                                                                                                                  
                                                                                                                                                     
                   Substitute each value into the original equation. 
                                                                                                                   
                                                                                                                 Substitute each value into the original equation. 
                                                                                                                   
                                                                                                                                                                              
                     
                   log9 (–4) is undefined and 2 is extraneous solution.                                            
                   Thus, x = 5.                                                                                  Since you can not have a log of 0, x =                      3 is the 
                               2                                                                                 solution. 
              18. log4 (2x + 1) = log4 (10x − 7)
                                                                                                                                   
                                                                                                                 SCIENCEThe equation for wind speed w, in 
                   SOLUTION:                                                                                     miles per hour, near the center of a tornado is w
                                                                                                                 = 93 log  d + 65, where d is the distance in 
                                                                                                                               10
                                                                                                                                                                   
                                                                                                                 miles that the tornado travels.
                                                                                                            20. Write this equation in exponential form.
                                                                                                                 SOLUTION:  
                     
                    
                                                       
                     
                   Substitute each value into the original equation.
                     
                                                                                
                                                                                                            21. In May of 1999, a tornado devastated Oklahoma City
                                                                                                                 with the fastest wind speed ever recorded. If the 
                                                                                                                 tornado traveled 525 miles, estimate the wind speed 
                   Thus, x = 1 or 4.                                                                             near the center of the tornado. 
                             2                                                                                   SOLUTION:  
              19. log7 (x − 4) = log7 (− x + 2)                                                                  Substitute 525 for d in the equation and simplify.
                                                                                                                   
                   SOLUTION:                                                                                                                     
                                                                
                                                                                                                 Solve each inequality.
                                                                                                                                   
                                                                                                            22. log6 x < −3
                     
                    
                                                                                                                 SOLUTION:  
                                                                                                                                     
                     
                   Substitute each value into the original equation. 
                     
                                                                                                                 The solution set is                                  .
                                                                                                                              
                   Since you can not have a log of 0, x =                      3 is the                     23. log4 x ≥ 4
                   solution. 
                                                                                                                 SOLUTION:  
                                                                                                                                     
                   SCIENCEThe equation for wind speed w, in 
                   miles per hour, near the center of a tornado is w
                   = 93 log  d + 65, where d is the distance in 
                                 10                                                                                
                                                                     
                   miles that the tornado travels.
              20. Write this equation in exponential form.                                                                                             
                                                                                                                 The solution set is {x | x ≥ 256}.
                   SOLUTION:                                                                                                       
                                                                                                            24. log3 x ≥ −4
                                                                                                                 SOLUTION:  
                                                                                                                                    
              21. In May of 1999, a tornado devastated Oklahoma City                                             The solution set is                          .
                   with the fastest wind speed ever recorded. If the 
                   tornado traveled 525 miles, estimate the wind speed                                                             
                   near the center of the tornado.                                                          25. log2 x ≤ −2
                                                                                                                 SOLUTION:  
                   SOLUTION:  
                   Substitute 525 for d in the equation and simplify.                                                               
                     
                                                   
                   Solve each inequality.                                                                        The solution set is                              .
                                     
              22. log6 x < −3
                                                                                                                              
                                                                                                            26. log     x > 2
                   SOLUTION:                                                                                          5
                                       
                                                                                                                 SOLUTION:  
                                                                                                                                   
                   The solution set is                                  .                                          
                                                                                                                 The solution set is                         .
                                
              23. log     x ≥ 4                                                                                                    
                        4                                                                                   27. log7 x < −1
                   SOLUTION:  
                                                                                                                 SOLUTION:  
                                                                                                                                   
                     
                                                         
                   The solution set is {x | x ≥ 256}.                                                            The solution set is                              .
                                     
              24. log3 x ≥ −4
                                                                                                                                                           
                                                                                                            28. log     (4x − 6) > log (2x + 8)
                   SOLUTION:                                                                                          2                     2
                                      
                                                                                                                 SOLUTION:  
                                                                                                                                                            
                   The solution set is                          .
                                                                                                                   
                                                                                                                 The solution set is                        .
              25. log2 x ≤ −2
                                                                                                                                                           
                                                                                                            29. log     (x + 2) ≥  log        (6x − 3)
                   SOLUTION:                                                                                          7                    7
                                      
                                                                                                                 SOLUTION:  
                                                                                                                                                          
                   The solution set is                              .
                                                                                                                   
                                                                                                                 Exclude all values of x for which 
              26. log5 x > 2                                                                                                                        
                                                                                                                   
                   SOLUTION:                                                                                                                               
                                                                                                                 So, 
                                                                                                                   
                                                                                                                 The solution set is                             . 
                   The solution set is                         .
                                                                                                                                                           
                                                                                                            30. log3 (7x – 6) < log3 (4x + 9)
              27. log7 x < −1
                                                                                                                 SOLUTION:  
                   SOLUTION:                                                                                                                                
                                     
                                                                                                                   
                   The solution set is                              .                                            Exclude all values of x for which 
                                                                                                                                                      
                                                                                                                   
              28. log2 (4x − 6) > log2 (2x + 8)                                                                                                             
                                                                                                                 So, 
                   SOLUTION:                                                                                       
                                                                                                                                                                    
                                                                                                                 The solution set is                              .
                                                                                                                                                             
                                                                                                            31. log5 (12x + 5) ≤ log5 (8x + 9)
                     
                   The solution set is                        .                                                  SOLUTION:  
                                                                                                                                                             
                                                             
              29. log7 (x + 2) ≥  log7 (6x − 3)
                   SOLUTION:  
                                                                                                                   
                                                                                                                 Exclude all values of x for which 
                                                                                                                                                       
                                                                                                                   
                                                                                                                 So,                                            
                   Exclude all values of x for which 
                                                                                                                   
                                                                                                                                                                        
                   So,                                                                                           The solution set is                                  .
                                                                                                                                                              
                                                                                                            32. log11 (3x − 24) ≥ log11 (−5x − 8)
                   The solution set is                             .
                                                                                                                 SOLUTION:  
                                                             
              30. log3 (7x – 6) < log3 (4x + 9)
                   SOLUTION:  
                                                                                                                   
                                                                                                                                                              
                                                                                                                 The solution set is {x | x ≥ 2}.
                                                                                                                                                         
                                                                                                            33. log9 (9x + 4) ≤ log9 (11x − 12)
                     
                   Exclude all values of x for which                                                             SOLUTION:  
                                                        
                     
                   So,                                        
                                                                                                                   
                                                                                                                                                       
                   The solution set is                              .                                            The solution set is {x | x ≥ 8}.
                                                                                                                                                  
                                                                                                            34. CCSS MODELINGThe magnitude of an 
                                                                                                                 earthquake is measured on a logarithmic scale called 
              31. log5 (12x + 5) ≤ log5 (8x + 9)                                                                                                                                               
                                                                                                                 the Richter scale. The magnitude M is given by M =
                   SOLUTION:                                                                                     log10 x, where x represents the amplitude of the 
                                                                                                                 seismic wave causing ground motion. 
                                                                                                                 a. How many times as great is the amplitude caused 
                                                                                                                 by an earthquake with a Richter scale rating of 8 as 
                                                                                                                 an aftershock with a Richter scale rating of 5? 
                                                                                                                 b. In 1906, San Francisco was almost completely 
                   Exclude all values of x for which                                                             destroyed by a 7.8 magnitude earthquake. In 1911, 
                                                                                                                 an earthquake estimated at magnitude 8.1 occurred 
                                                                                                                 along the New Madrid fault in the Mississippi River 
                                                                                                                 Valley. How many times greater was the New 
                   So,                                                                                           Madrid earthquake than the San Francisco 
                                                                                                                 earthquake? 
                     
                                                                                                                 SOLUTION:  
                   The solution set is                                  .                                        a. 
                                                                                                                 The amplitude of the seismic wave with a Richter 
                                                                                                                 scale rating of 8 and 5 are 108 and 105 respectively. 
                                                                                                                               8          5
              32. log11 (3x − 24) ≥ log11 (−5x − 8)                                                              Divide 10  by 10 . 
                                                                                                                   
                   SOLUTION:                                                                                                       
                                                                                                                   
                                                                                                                 The scale rating of 8 is 103 or 1000 times greater 
                                                                                                                 than the scale rating of 5. 
                                                                
                   The solution set is {x | x ≥ 2}.                                                                
                                                                                                                 b. 
              33. log9 (9x + 4) ≤ log9 (11x − 12)                                                                The amplitudes of San Francisco earthquake and 
                                                                                                                 New Madrid earthquake were 107.8 and 108.1 
                   SOLUTION:                                                                                     respectively. 
                                                                                                                   
                                                                                                                 Divide 108.1 by 107.8. 
                                                                                                                   
                                                                                                                  
                                                                                                                                        
                     
                                                         
                   The solution set is {x | x ≥ 8}.
                                                                                                                   
                                                                                                                  
              34. CCSS MODELINGThe magnitude of an                                                                                                                      0.3
                   earthquake is measured on a logarithmic scale called                                          The New Madrid earthquake was 10                           or about 2 
                                                                                                                 times greater than the San Francisco earthquake. 
                   the Richter scale. The magnitude M is given by M =
                   log10 x, where x represents the amplitude of the                                                            
                                                                                                                 MUSICThe first key on a piano keyboard 
                   seismic wave causing ground motion.                                                      35. 
                   a. How many times as great is the amplitude caused                                            corresponds to a pitch with a frequency of 27.5 
                   by an earthquake with a Richter scale rating of 8 as                                          cycles per second. With every successive key, going 
                   an aftershock with a Richter scale rating of 5?                                               up the black and white keys, the pitch multiplies by a 
                                                                                                                 constant. The formula for the frequency of the pitch 
                   b. In 1906, San Francisco was almost completely                                               sounded when the nth note up the keyboard is played
                   destroyed by a 7.8 magnitude earthquake. In 1911,                                             is given by                                     
                   an earthquake estimated at magnitude 8.1 occurred 
                   along the New Madrid fault in the Mississippi River                                           a. A note has a frequency of 220 cycles per second. 
                   Valley. How many times greater was the New                                                    How many notes up the piano keyboard is this? 
                   Madrid earthquake than the San Francisco                                                      b. Another pitch on the keyboard has a frequency of 
                   earthquake?                                                                                   880 cycles per second. After how many notes up the
                                                                                                                 keyboard will this be found? 
                   SOLUTION:  
                   a. 
                   The amplitude of the seismic wave with a Richter                                              SOLUTION:  
                   scale rating of 8 and 5 are 108 and 105 respectively.                                         a. 
                                 8          5                                                                    Substitute 220 for f  in the formula and solve for n.
                                                                                                                   
                   Divide 10  by 10 .                                                                             
                                                                                                                                                
                                     
                     
                   The scale rating of 8 is 103 or 1000 times greater                                            b. 
                   than the scale rating of 5.                                                                   Substitute 880 for f  in the formula and solve for n.
                                                                                                                   
                   b.                                                                                             
                   The amplitudes of San Francisco earthquake and                                                                               
                   New Madrid earthquake were 107.8 and 108.1 
                   respectively. 
                     
                   Divide 108.1 by 107.8. 
                     
                                                                                                                                                                             
                                                                                                                 MULTIPLE REPRESENTATIONSIn this 
                                                                                                            36.                                                                   
                                                                                                                 problem, you will explore the graphs shown:
                                                                                                                                                         
                                                                                                                 y = log4 x and 
                                                                                                                   
                                                                                                                  
                    
                   The New Madrid earthquake was 100.3 or about 2 
                   times greater than the San Francisco earthquake. 
                                 
                   MUSICThe first key on a piano keyboard 
              35. 
                   corresponds to a pitch with a frequency of 27.5 
                   cycles per second. With every successive key, going 
                   up the black and white keys, the pitch multiplies by a                                                                             
                   constant. The formula for the frequency of the pitch                                            
                   sounded when the nth note up the keyboard is played                                            
                                                                                                                 a. ANALYTICAL
                                                                                                                                               How do the shapes of the 
                   is given by                                                                                   graphs compare? How do the asymptotes and the x-
                                                                                                                 intercepts of the graphs compare? 
                   a. A note has a frequency of 220 cycles per second.                                           b. VERBAL
                                                                                                                                      Describe the relationship between the 
                   How many notes up the piano keyboard is this?                                                 graphs.
                   b.                                                                                                
                       Another pitch on the keyboard has a frequency of                                          c. GRAPHICAL
                                                                                                                                             Use what you know about 
                   880 cycles per second. After how many notes up the                                            transformations of graphs to compare and contrast 
                   keyboard will this be found?                                                                                                                                               
                                                                                                                 the graph of each function and the graph of y = log4
                                                                                                                 x. 
                   SOLUTION:                                                                                       
                   a.                                                                                             
                                                                                                                                        
                   Substitute 220 for f  in the formula and solve for n.                                         1. y = log4 x + 2
                     
                                                                                                                                             
                                                                                                                 2. y = log4 (x + 2)
                                                                                                                                      
                                                                                                                 3. y = 3 log4 x
                                                                                                                   
                                                                                                                  
                                                                                                                 d. ANALYTICAL
                                                                                                                                                Describe the relationship 
                                                                                                                                                                      
                                                                                                                 between y = log x and y =  1(log x). What are a 
                                                                                                                                         4                −        4
                   b.                                                                                            reasonable domain and range for each function? 
                                                                                                                     
                                                                                                                 e.ANALYTICAL
                   Substitute 880 for f  in the formula and solve for n.                                                                        Write an equation for a function 
                                                                                                                 for which the graph is the graph of y = log3 x 
                                                                                                                                                                      
                                                                                                                 translated 4 units left and 1 unit up.
                                                                                                                 SOLUTION:  
                                                                                                                 a. 
                                                                                                                 The shapes of the graphs are the same. The 
                                                                                                                 asymptote for each graph is the y-axis and the x-
                                                                                                                 intercept for each graph is 1. 
                                                                                                                 b. 
                   MULTIPLE REPRESENTATIONSIn this 
              36.                                                                                                The graphs are reflections of each other over the x-
                   problem, you will explore the graphs shown:                                                   axis. 
                                                           
                   y = log4 x and                                                                                    
                                                                                                                 c.
                                                                                                                 1. The second graph is the same as the first, except it
                                                                                                                 is shifted horizontally to the left 2 units. 
                                                                                                                   
                                                                                                                  
                                                        
                                                                                                                                                                          
                   a. ANALYTICAL                                                                                       
                                                 How do the shapes of the                                          
                   graphs compare? How do the asymptotes and the x-                                               
                                                                                                                             2, 8] scl: 1 by [ 5, 5] scl: 1 
                   intercepts of the graphs compare?                                                                     [−                       −
                                                                                                                   
                   b. VERBAL                                                                                      
                                        Describe the relationship between the                                        
                   graphs.                                                                                       2. The second graph is the same as the first, except it
                                                                                                                 is shifted vertically up 2 units. 
                   c. GRAPHICAL
                                               Use what you know about                                             
                   transformations of graphs to compare and contrast                                              
                                                                                                
                   the graph of each function and the graph of y = log4
                   x. 
                     
                    
                                          
                   1. y = log4 x + 2
                                               
                   2. y = log4 (x + 2)
                                        
                   3. y = 3 log x                                                                                                                                       
                                    4                                                                                  
                                                                                                                   
                                                                                                                  
                                                                                                                                                                   
                                                                                                                            4, 8] scl: 1 by [ 5, 5] scl: 1
                   d. ANALYTICAL                                                                                        [−                       −
                                                  Describe the relationship                                        
                                                                                                                  
                   between y = log x and y =  1(log x). What are a 
                                           4                −        4                                           3. Each point on the second graph has a y-coordinate
                   reasonable domain and range for each function?                                                3 times that of the corresponding point on the first 
                       
                   e.ANALYTICAL
                                                  Write an equation for a function                               graph. 
                   for which the graph is the graph of y = log3 x                                                  
                                                                                                                  
                   translated 4 units left and 1 unit up.
                   SOLUTION:  
                   a. 
                   The shapes of the graphs are the same. The 
                   asymptote for each graph is the y-axis and the x-
                   intercept for each graph is 1.                                                                                                                      
                   b.                                                                                                  
                                                                                                                   
                   The graphs are reflections of each other over the x-                                           
                   axis.                                                                                                   2, 8] scl: 1 by [ 5, 5] scl: 1 
                                                                                                                       [−                       −
                                                                                                                     
                   c.                                                                                            d.
                   1. The second graph is the same as the first, except it                                       The graphs are reflections of each other over the x-
                   is shifted horizontally to the left 2 units.                                                  axis. 
                                                                                                                                                                            
                                                                                                                 D={x|x>0}; R={all real numbers}
                    
                                                                                                                 e.                                   where h is the horizontal 
                                                                                                                 shift and k is the vertical shift.  Since there is a 
                                                                                                                 horizontal shift of 4 and vertical shift of 1, h = 4 and 
                                                                                                                 k = 1. 
                                                                                                                 y = log3 (x + 4) + 1 
                                                                                                                                
                                                                                                                 SOUNDThe relationship between the intensity of 
                                                                                                            37. 
                                                                                                                 sound I and the number of decibels   
                                                                                                                                                                      β
                    
                               2, 8] scl: 1 by [ 5, 5] scl: 1 
                           [−                       −                                                            is                               , where I is the intensity of 
                     
                    
                   2. The second graph is the same as the first, except it                                       sound in watts per square meter. 
                                                                                                                   
                   is shifted vertically up 2 units.                                                              
                                                                                                                     
                                                                                                                 a. Find the number of decibels of a sound with an 
                                                                                                                 intensity of 1 watt per square meter. 
                                                                                                                 b. Find the number of decibels of sound with an 
                                                                                                                                      −2                                      
                                                                                                                 intensity of 10          watts per square meter.
                                                                                                                     
                                                                                                                 c. The intensity of the sound of 1 watt per square 
                                                                                                                 meter is 100 times as much as the intensity of 10−2 
                                                                                                                 watts per square meter. Why are the decibels of 
                                                                                                                 sound not 100 times as great? 
                     
                    
                              4, 8] scl: 1 by [ 5, 5] scl: 1                                                     SOLUTION:  
                          [−                       −                                                             a. 
                     
                                                                                                                 Substitute 1 for I in the given equation and solve for 
                   3. Each point on the second graph has a y-coordinate                                            . 
                   3 times that of the corresponding point on the first                                          β
                                                                                                                   
                   graph.                                                                                         
                                                                                                                                                
                    
                                                                                                                   
                                                                                                                  
                                                                                                                 b. 
                     
                                                                                                                                    −2
                             2, 8] scl: 1 by [ 5, 5] scl: 1                                                      Substitute 10          for I in the given equation and solve 
                         [−                       −
                   d.                                                                                            for  . 
                   The graphs are reflections of each other over the x-                                                β
                                                                                                                   
                   axis.                                                                                          
                                                                                                                                                
                   D={x|x>0}; R={all real numbers}
                   e.                                   where h is the horizontal 
                   shift and k is the vertical shift.  Since there is a 
                   horizontal shift of 4 and vertical shift of 1, h = 4 and 
                   k = 1. 
                   y = log3 (x + 4) + 1 
                                                                                                                   
                                                                                                                  
                   SOUNDThe relationship between the intensity of 
              37.                                                                                                    
                   sound I and the number of decibels                                                            c. Sample answer: The power of the logarithm only 
                                                                        β                                        changes by 2. The power is the answer to the 
                   is                               , where I is the intensity of                                logarithm. That 2 is multiplied by the 10 before the 
                                                                                                                 logarithm. So we expect the decibels to change by 
                   sound in watts per square meter.                                                              20. 
                     
                    
                                                                                                                 CCSS CRITIQUE  
                   a. Find the number of decibels of a sound with an                                        38.                                  Ryan and Heather are solving 
                                                                                                                 log  x    3. Is either of them correct? Explain your 
                   intensity of 1 watt per square meter.                                                              3    ≥−
                   b. Find the number of decibels of sound with an                                               reasoning. 
                                        −2                                                                         
                                                                                                                  
                   intensity of 10          watts per square meter.
                       
                   c. The intensity of the sound of 1 watt per square 
                   meter is 100 times as much as the intensity of 10−2 
                   watts per square meter. Why are the decibels of 
                   sound not 100 times as great? 
                   SOLUTION:                                                                                                                                        
                   a.                                                                                             
                   Substitute 1 for I in the given equation and solve for 
                     . 
                   β
                     
                    
                                                  
                                                                                                                                                                    
                                                                                                                 SOLUTION:  
                                                                                                                 Sample answer: Ryan; Heather did not need to 
                                                                                                                 switch the inequality symbol when raising to a 
                                                                                                                 negative power. 
                     
                                                                                                                                                                                              
                                                                                                                 CHALLENGEFind log 27 +log 27 +log                                    27 +
                   b.                                                                                       39.                                       3             9             27
                   Substitute 10−2 for I in the given equation and solve                                                                        
                                                                                                                 log81 27 + log243 27.
                   for  . 
                         β
                                                                                                                 SOLUTION:  
                    
                                                  
                     
                                                                                                                                         
                                                                                                                 REASONINGThe Property of Inequality for 
                                                                                                            40. 
                       
                   c. Sample answer: The power of the logarithm only                                                                                                                          
                                                                                                                 Logarithmic Functions states that when b > 1, logb x
                   changes by 2. The power is the answer to the                                                                                        
                   logarithm. That 2 is multiplied by the 10 before the                                          >logby if and only if x >y. What is the case for 
                                                                                                                                                                              
                   logarithm. So we expect the decibels to change by                                             when 0 < b < 1? Explain your reasoning.
                   20. 
                                                                                                                 SOLUTION:  
                                                                                                                                                                                        
                   CCSS CRITIQUE                                                                                 Sample answer: When 0 < b < 1, log x > log y if 
              38.                                  Ryan and Heather are solving                                                                                        b           b
                   log  x    3. Is either of them correct? Explain your 
                        3    ≥−                                                                                                        
                                                                                                                 and only if x  1, logb x
                                                                                                                 SOLUTION:  
                   >logby if and only if x >y. What is the case for                                              a. less than 
                                                                                
                   when 0 < b < 1? Explain your reasoning.                                                       b. less than 
                                                                                                                          
                                                                                                                 c. no
                   SOLUTION:                                                                                     d. infinitely many
                                                                                          
                   Sample answer: When 0 < b < 1, logb x > logby if 
                                         
                   and only if x 81
                                                                                                                 SOLUTION:  
                                                                                                                                   
              51. log6 216
                   SOLUTION:  
                   log  27                                                                                         4a + 6          a 
              52.                                                                                                4        16
                        3                                                                                   57.             ≤
                                                                                                                 SOLUTION:  
                   SOLUTION:  
                                                                                                                                     
                                  
              53. 
                   SOLUTION:                                                                                                          
                                                                                                            58. 
                                                                                                                 SOLUTION:  
                   log  2401 
              54.       7
                   SOLUTION:  
                   Solve each equation or inequality. Check your                                                                      
                                                                                                            59. 
                   solution.
                     2x + 3            
                   5        125
              55.             ≤
                                                                                                                 SOLUTION:  
                   SOLUTION:  
                                       
                     3x − 2           
                               
              56. 3           >81
                                                                                                                                   
                   SOLUTION:  
                                                                                                            60. 
                                     
                                                                                                                 SOLUTION:  
                     4a + 6          a 
                   4        16
              57.             ≤
                   SOLUTION:  
                                                                                                                 SHIPPING The height of a shipping cylinder is 4 
                                                                                                            61. 
                                                                                                                 feet more than the radius. If the volume of the 
                                                                                                                 cylinder is 5  cubic feet, how tall is it? Use the 
                                                                                                                                   π
                                                                                                                                       2     
                                                                                                                 formula V = πr h.
                                                                                                                 SOLUTION:  
                                                                                                                 Substitute 5  for V and r + 4 for h in the formula and
                                                                                                                                  π
                                                                                                                 simplify.
              58.                                                                                                  
                                                                                                                  
                   SOLUTION:                                                                                                                            
                                                                                                                   
                                                                                                                  
                                                                                                                 The equation has one real root r = 1. 
                                                                                                                 Thus, the height of the shipping cylinder is 1 + 4 = 5 
                                                                                                                      
                                                                                                                 ft.
                                                                                                                                                     
                                                                                                                 NUMBER THEORYTwo complex conjugate 
                                                                                                            62. 
                                                                                                                 numbers have a sum of 12 and a product of 40. Find 
                                                                                                                 the two numbers 
              59. 
                                                                                                                 SOLUTION:  
                   SOLUTION:  
                                                                                                                 The equations that represent the situation are:
                                                                                                                   
                                                                                                                  
                                                                                                                                                             
                                                                                                                   
                                                                                                                  
                                                                                                                 Solve equation (1). 
                                                                                                                   
                                                                                                                  
                                                                                                                                                   
                                     
              60. 
                   SOLUTION:                                                                                       
                                                                                                                  
                                                                                                                 Solve equation (2). 
                                                                                                                   
                                                                                                                  
                                                                                                                                                                              
                                                                                                                   
                                                                                                                  
                   SHIPPING The height of a shipping cylinder is 4                                               Thus, the two numbers are 6 + 2i and 6   2i. 
              61.                                                                                                                                                           –
                   feet more than the radius. If the volume of the                                               Simplify. Assume that no variable equals zero.
                   cylinder is 5  cubic feet, how tall is it? Use the 
                                     π                                                                             5      3 
                                         2                                                                       x    x
                                                                                                            63.       ·
                   formula V = πr h.
                                                                                                                 SOLUTION:  
                   SOLUTION:                                                                                                          
                   Substitute 5  for V and r + 4 for h in the formula and
                                    π
                   simplify.
                     
                    
                                                                                                                 a2   a6 
                                                                                                            64.       ·
                                                                                                                 SOLUTION:  
                                                                                                                                       
                   The equation has one real root r = 1. 
                   Thus, the height of the shipping cylinder is 1 + 4 = 5 
                        
                   ft.
                                                                                                                 (2p2n)3 
                                                                                                            65. 
                   NUMBER THEORYTwo complex conjugate 
              62. 
                   numbers have a sum of 12 and a product of 40. Find 
                   the two numbers                                                                               SOLUTION:  
                   SOLUTION:  
                   The equations that represent the situation are:                                               (3b3c2)2 
                                                                                                            66. 
                    
                                                               
                                                                                                                 SOLUTION:  
                     
                    
                   Solve equation (1).                                                                                     
                                                                                                            67. 
                    
                                                     
                                                                                                                 SOLUTION:  
                                                                                                                                     
                     
                    
                   Solve equation (2).                                                                                       
                     
                    
                                                                                                            68. 
                                                                                                                 SOLUTION:  
                                                                                                                                  
                     
                    
                   Thus, the two numbers are 6 + 2i and 6   2i. 
                                                                              –
                   Simplify. Assume that no variable equals zero.
                   x5   x3 
              63.       ·
                   SOLUTION:  
                                        
                   a2   a6 
              64.       ·
                   SOLUTION:  
                                         
                   (2p2n)3 
              65. 
                   SOLUTION:  
                   (3b3c2)2 
              66. 
                   SOLUTION:  
                             
              67. 
                   SOLUTION:  
                                       
                               
              68. 
                   SOLUTION:  
                                    
                                                                                                                 Solve each equation.
                                                                                                                                  
                                                                                                              1. 
                                                                                                                 SOLUTION:  
                                                                                                                                        
                                                                                                                                   
                                                                                                              2. 
                                                                                                                 SOLUTION:  
                                                                                                                                        
                                                                                                                                                                          2                    
                                                                                                                 MULTIPLE CHOICE
                                                                                                              3.                                       Solve log5 (x − 10) = log5
                                                                                                                       
                                                                                                                 3x.
                                                                                                                 A 10 
                                                                                                                 B   
                                                                                                                     2
                                                                                                                 C   
                                                                                                                     5
                                                                                                                 D 2, 5 
                                                                                                                 SOLUTION:  
                                                                                                                                                     
                   Solve each equation.
                                                                                                                   
                1. 
                                                                                                                                                       
                   SOLUTION:  
                                                                                                                   
                                                                                                                 Substitute each value into the original equation. 
                                                                                                                   
                                                                                                                                                                              
                                                                                                                   
                                                                                                                 The domain of a logarithmic function cannot be 0, so 
                                                                                                                 log5 (–6) is undefined and –2 is an extraneous 
                                                                                                                 solution. 
                2.                                                                                                 
                                                                                                                 C is the correct option. 
                   SOLUTION:  
                                                                                                                 Solve each inequality.
                                                                                                                              
                                                                                                              4. log5 x > 3
                                                                                                                 SOLUTION:  
                                                                                                                                    
                                                                                                                   
                                                                                                                                                          
                                                                            2                                    Thus, solution set is  {x | x > 125}.
                   MULTIPLE CHOICE
                3.                                       Solve log5 (x − 10) = log5
                         
                   3x.                                                                                                             
                   A 10                                                                                       5. log8 x ≤ −2
                   B   
                       2
                   C                                                                                             SOLUTION:  
                       5                                                                                                           
                   D 2, 5 
                   SOLUTION:  
                                                       
                                                                                                                   
                                                                                                                 Thus, solution set is                                 .
                                                                                                                                                           
                                                                                                              6. log4 (2x + 5) ≤ log4 (4x − 3)
                                                                                                                 SOLUTION:  
                                                                                                                                                            
                   Substitute each value into the original equation. 
                     
                                                                                
                                                                                                                   
                                                                                                                                                         
                                                                                                                 Thus, solution set is {x | x ≥ 4}.
                                                                                                                                                      
                   The domain of a logarithmic function cannot be 0, so                                       7. log8 (2x) > log8 (6x − 8)
                   log5 (–6) is undefined and –2 is an extraneous 
                   solution.                                                                                     SOLUTION:  
                                                                                                                                                      
                   C is the correct option. 
                   Solve each inequality.
                                
                4. log5 x > 3                                                                                      
                                                                                                                 Exclude all values of x for which 
                   SOLUTION:                                                                                                                     
                                                                                                                   
                                                                                                                 So,                                     
                                                                                                                   
                                                            
                   Thus, solution set is  {x | x > 125}.                                                                                                               
                                                                                                                 Thus, solution set is                               .
                                     
                5. log8 x ≤ −2
                                                                                                                 CCSS STRUCTURE  Solve each equation.
                   SOLUTION:                                                                                                       
                                                                                                              8. 
                                                                                                                 SOLUTION:  
                                                                                                                                        
                     
                   Thus, solution set is                                 .
                                                             
                6. log4 (2x + 5) ≤ log4 (4x − 3)
                   SOLUTION:  
                                                                                                                                   
                                                                                                              9. 
                                                                                                                 SOLUTION:  
                                                                                                                                         
                                                           
                   Thus, solution set is {x | x ≥ 4}.
                                                        
                7. log8 (2x) > log8 (6x − 8)
                   SOLUTION:  
                                                        
                                                                                                                                  
                                                                                                            10. 
                     
                   Exclude all values of x for which                                                             SOLUTION:  
                                                                                                                                     
                     
                   So,                                     
                     
                   Thus, solution set is                               . 
                   CCSS STRUCTURE  Solve each equation.
                                                                                                                                    
                8. 
                                                                                                            11. 
                   SOLUTION:  
                                                                                                                 SOLUTION:  
                                                                                                                                       
                                                                                                                                    
                                                                                                            12. 
                                     
                9. 
                                                                                                                 SOLUTION:  
                                                                                                                                     
                   SOLUTION:  
                                           
                                                                                                                                    
                                                                                                            13. 
              10. 
                                                                                                                 SOLUTION:  
                   SOLUTION:                                                                                                         
                                       
            7-4 Solving Logarithmic Equations and Inequalities
                                                                                                 
                                                                                                                                                  2        
                                                                                                            14. log3 (3x + 8) = log3 (x + x)
              11. 
                                                                                                                 SOLUTION:  
                   SOLUTION:                                                                                                                                  
                                         
                                                                                                                   
                                                                                                                                                   
                                                                                                                   
              12. 
                                                                                                                 Substitute each value into the original equation.
                                                                                                                   
                   SOLUTION:                                                                                                                                                  
                                       
                                                                                                                   
                                                                                                                 Thus, x = –2 or 4. 
                                                                                                                             2                              
                                                                                                            15. log12 (x − 7) = log12 (x + 5)
                                                                                                                 SOLUTION:  
                                                                                                                                                            
                                      
              13. 
                   SOLUTION:                                                                                       
                                                                                                                  
                                                                                                                                                   
                                                                                                                   
                                                                                                                 Substitute each value into the original equation.
                                                                                                                   
                                                                                                                                                                              
                                                                                                                   
                                                    2                                                            Thus, x = –3 or 4. 
              14. log3 (3x + 8) = log3 (x + x)
                                                                                                                           2                           
                                                                                                            16. log6 (x − 6x) = log6 (−8)
                   SOLUTION:  
                                                                                                                 SOLUTION:  
                                                                                                                                                         
                     
            eSolutions Manual - Powered by Cognero                                                                                                                                      Page3
                                                                                                                  
                                                                                                                                                 
                     
                   Substitute each value into the original equation.                                               
                                                                                                                 Substitute each value into the original equation. 
                                                                                                                   
                                                                                                                                                                              
                     
                   Thus, x = –2 or 4.                                                                              
                               2                                                                                 log  (–8) is undefined, so 4 and 2 are extraneous 
              15. log12 (x − 7) = log12 (x + 5)                                                                       6
                                                                                                                 solutions. 
                   SOLUTION:                                                                                     Thus, no solution. 
                                                                                                                           2                                   
                                                                                                                                                         
                                                                                                            17. log9 (x − 4x) = log9 (3x − 10)
                                                                                                                 SOLUTION:  
                                                                                                                                                               
                     
                    
                                                     
                                                                                                                   
                   Substitute each value into the original equation.                                              
                                                                                                                                                 
                                                                                
                                                                                                                   
                                                                                                                 Substitute each value into the original equation. 
                                                                                                                   
                   Thus, x = –3 or 4.                                                                                                                                         
                             2                           
              16. log6 (x − 6x) = log6 (−8)
                                                                                                                   
                   SOLUTION:                                                                                     log  (–4) is undefined and 2 is extraneous solution.
                                                                                                                      9
                                                                                                                 Thus, x = 5. 
                                                                                                                             2                                 
                                                                                                            18. log4 (2x + 1) = log4 (10x − 7)
                                                                                                                 SOLUTION:  
                     
                                                                                                                                                               
                                                   
                     
                   Substitute each value into the original equation. 
                                                                                                                   
                                                                                                                  
                                                                                                                                                     
                                                                                                                   
                   log  (–8) is undefined, so 4 and 2 are extraneous                                             Substitute each value into the original equation.
                        6                                                                                          
                   solutions.                                                                                                                                                 
                   Thus, no solution. 
                             2                                                                                     
              17. log9 (x − 4x) = log9 (3x − 10)                                                                 Thus, x = 1 or 4. 
                   SOLUTION:                                                                                               2                                
                                                                                                            19. log7 (x − 4) = log7 (− x + 2)
                                                                                                                 SOLUTION:  
                                                                                                                                                              
                     
                    
                                                   
                                                                                                                   
                                                                                                                  
                   Substitute each value into the original equation.                                                                                 
                     
                                                                                                                   
                                                                                                                 Substitute each value into the original equation. 
                                                                                                                   
                                                                                                                                                                              
                   log9 (–4) is undefined and 2 is extraneous solution.
                   Thus, x = 5.                                                                                    
                               2                                                                                 Since you can not have a log of 0, x =                      3 is the 
              18. log4 (2x + 1) = log4 (10x − 7)                                                                 solution. 
                   SOLUTION:                                                                                                       
                                                                                                                 SCIENCEThe equation for wind speed w, in 
                                                                                                                 miles per hour, near the center of a tornado is w
                                                                                                                 = 93 log  d + 65, where d is the distance in 
                                                                                                                               10
                                                                                                                                                                   
                                                                                                                 miles that the tornado travels.
                                                                                                            20. Write this equation in exponential form.
                                                                                                                 SOLUTION:  
                    
                                                       
                     
                   Substitute each value into the original equation.
                     
                                                                                
                                                                                                            21. In May of 1999, a tornado devastated Oklahoma City
                                                                                                                 with the fastest wind speed ever recorded. If the 
                   Thus, x = 1 or 4.                                                                             tornado traveled 525 miles, estimate the wind speed 
                             2                                                                                   near the center of the tornado. 
                                                         
              19. log7 (x − 4) = log7 (− x + 2)
                                                                                                                 SOLUTION:  
                                                                                                                 Substitute 525 for d in the equation and simplify.
                   SOLUTION:                                                                                       
                                                                                                                                                 
                                                                                                                 Solve each inequality.
                                                                                                                                   
                                                                                                            22. log6 x < −3
                    
                                                       
                                                                                                                 SOLUTION:  
                                                                                                                                     
                     
                   Substitute each value into the original equation. 
                     
                                                                                
                                                                                                                 The solution set is                                  .
                     
                   Since you can not have a log of 0, x =                      3 is the                                       
                   solution.                                                                                23. log4 x ≥ 4
                                                                                                                 SOLUTION:  
                   SCIENCEThe equation for wind speed w, in                                                                          
                   miles per hour, near the center of a tornado is w
                   = 93 log  d + 65, where d is the distance in 
                                 10
                                                                     
                   miles that the tornado travels.
              20. Write this equation in exponential form.                                                         
                                                                                                                                                       
                                                                                                                 The solution set is {x | x ≥ 256}.
                   SOLUTION:  
                                                                                                                                   
                                                                                                            24. log3 x ≥ −4
                                                                                                                 SOLUTION:  
                                                                                                                                    
              21. In May of 1999, a tornado devastated Oklahoma City
                   with the fastest wind speed ever recorded. If the                                             The solution set is                          .
                   tornado traveled 525 miles, estimate the wind speed 
                   near the center of the tornado.                                                                                 
                                                                                                            25. log2 x ≤ −2
                   SOLUTION:  
                   Substitute 525 for d in the equation and simplify.                                            SOLUTION:  
                                                                                                                                    
                                                   
                   Solve each inequality.
                                     
              22. log6 x < −3                                                                                    The solution set is                              .
                   SOLUTION:                                                                                                  
                                                                                                            26. log5 x > 2
                                                                                                                 SOLUTION:  
                                                                                                                                   
                   The solution set is                                  .                                          
                                                                                                                 The solution set is                         .
              23. log4 x ≥ 4
                                                                                                                                   
                                                                                                            27. log7 x < −1
                   SOLUTION:  
                                       
                                                                                                                 SOLUTION:  
                                                                                                                                   
                     
                                                         
                   The solution set is {x | x ≥ 256}.
                                     
              24. log3 x ≥ −4                                                                                    The solution set is                              .
                   SOLUTION:                                                                                                                               
                                                                                                            28. log2 (4x − 6) > log2 (2x + 8)
                                                                                                                 SOLUTION:  
                                                                                                                                                            
                   The solution set is                          .
                                                                                                                   
              25. log2 x ≤ −2                                                                                    The solution set is                        .
                   SOLUTION:                                                                                                                               
                                                                                                            29. log7 (x + 2) ≥  log7 (6x − 3)
                                                                                                                 SOLUTION:  
                                                                                                                                                          
                   The solution set is                              .
                                                                                                                   
              26. log5 x > 2                                                                                     Exclude all values of x for which 
                                                                                                                                                    
                   SOLUTION:                                                                                       
                                                                                                                                                           
                                                                                                                 So, 
                                                                                                                   
                                                                                                                 The solution set is                             . 
                   The solution set is                         .
                                     
              27. log7 x < −1                                                                                                                              
                                                                                                            30. log3 (7x – 6) < log3 (4x + 9)
                   SOLUTION:  
                                                                                                                 SOLUTION:  
                                                                                                                                                            
                   The solution set is                              .                                              
                                                                                                                 Exclude all values of x for which 
                                                                                                                                                      
              28. log2 (4x − 6) > log2 (2x + 8)                                                                    
                                                                                                                                                            
                   SOLUTION:                                                                                     So, 
                                                                                                                   
                                                                                                                 The solution set is                              . 
                                                                                                                                                             
                                                                                                            31. log5 (12x + 5) ≤ log5 (8x + 9)
                   The solution set is                        .
                                                                                                                 SOLUTION:  
                                                                                                                                                             
              29. log7 (x + 2) ≥  log7 (6x − 3)
                   SOLUTION:  
                                                            
                                                                                                                   
                                                                                                                 Exclude all values of x for which 
                                                                                                                                                       
                                                                                                                   
                   Exclude all values of x for which                                                             So,                                            
                                                      
                                                                                                                   
                   So,                                                                                                                                                  
                                                                                                                 The solution set is                                  .
                     
                                                                                                                                                              
                   The solution set is                             .                                        32. log11 (3x − 24) ≥ log11 (−5x − 8)
                                                                                                                 SOLUTION:  
                                                             
              30. log3 (7x – 6) < log3 (4x + 9)
                   SOLUTION:  
                                                              
                                                                                                                   
                                                                                                                                                              
                                                                                                                 The solution set is {x | x ≥ 2}.
                                                                                                                                                         
                                                                                                            33. log9 (9x + 4) ≤ log9 (11x − 12)
                   Exclude all values of x for which 
                                                                                                                 SOLUTION:  
                     
                   So,                                        
                     
                   The solution set is                              .                                              
                                                                                                                                                       
                                                                                                                 The solution set is {x | x ≥ 8}.
                                                                                                                                                  
                                                                                                            34. CCSS MODELINGThe magnitude of an 
              31. log5 (12x + 5) ≤ log5 (8x + 9)                                                                 earthquake is measured on a logarithmic scale called 
                   SOLUTION:                                                                                                                                                                   
                                                                                                                 the Richter scale. The magnitude M is given by M =
                                                                                                                 log10 x, where x represents the amplitude of the 
                                                                                                                 seismic wave causing ground motion. 
                                                                                                                 a. How many times as great is the amplitude caused 
                                                                                                                 by an earthquake with a Richter scale rating of 8 as 
                                                                                                                 an aftershock with a Richter scale rating of 5? 
                   Exclude all values of x for which                                                             b. In 1906, San Francisco was almost completely 
                                                                                                                 destroyed by a 7.8 magnitude earthquake. In 1911, 
                                                                                                                 an earthquake estimated at magnitude 8.1 occurred 
                                                                                                                 along the New Madrid fault in the Mississippi River 
                   So,                                                                                           Valley. How many times greater was the New 
                                                                                                                 Madrid earthquake than the San Francisco 
                                                                                                                 earthquake? 
                                                                          
                   The solution set is                                  .                                        SOLUTION:  
                                                                                                                 a. 
                                                                                                                 The amplitude of the seismic wave with a Richter 
                                                                                                                                                            8           5                     
              32. log11 (3x − 24) ≥ log11 (−5x − 8)                                                              scale rating of 8 and 5 are 10  and 10  respectively.
                                                                                                                 Divide 108 by 105. 
                   SOLUTION:                                                                                       
                                                                                                                                   
                                                                                                                   
                                                                                                                 The scale rating of 8 is 103 or 1000 times greater 
                                                                
                   The solution set is {x | x ≥ 2}.                                                              than the scale rating of 5. 
                                                                                                                   
              33. log9 (9x + 4) ≤ log9 (11x − 12)                                                                b. 
                                                                                                                 The amplitudes of San Francisco earthquake and 
                   SOLUTION:                                                                                     New Madrid earthquake were 107.8 and 108.1 
                                                                                                                 respectively. 
                                                                                                                   
                                                                                                                 Divide 108.1 by 107.8. 
                                                                                                                   
                                                                                                                  
                                                                                                                                        
                                                         
                   The solution set is {x | x ≥ 8}.
                                                    
              34. CCSS MODELINGThe magnitude of an                                                                 
                   earthquake is measured on a logarithmic scale called                                           
                                                                                                                 The New Madrid earthquake was 100.3 or about 2 
                   the Richter scale. The magnitude M is given by M =                                            times greater than the San Francisco earthquake. 
                   log10 x, where x represents the amplitude of the 
                   seismic wave causing ground motion.                                                                         
                                                                                                                 MUSICThe first key on a piano keyboard 
                   a. How many times as great is the amplitude caused                                       35. 
                   by an earthquake with a Richter scale rating of 8 as                                          corresponds to a pitch with a frequency of 27.5 
                   an aftershock with a Richter scale rating of 5?                                               cycles per second. With every successive key, going 
                   b. In 1906, San Francisco was almost completely                                               up the black and white keys, the pitch multiplies by a 
                   destroyed by a 7.8 magnitude earthquake. In 1911,                                             constant. The formula for the frequency of the pitch 
                   an earthquake estimated at magnitude 8.1 occurred                                             sounded when the nth note up the keyboard is played
                   along the New Madrid fault in the Mississippi River                                           is given by                                     
                   Valley. How many times greater was the New                                                    a. A note has a frequency of 220 cycles per second. 
                   Madrid earthquake than the San Francisco                                                      How many notes up the piano keyboard is this? 
                   earthquake?                                                                                   b. Another pitch on the keyboard has a frequency of 
                   SOLUTION:                                                                                     880 cycles per second. After how many notes up the
                   a.                                                                                            keyboard will this be found? 
                   The amplitude of the seismic wave with a Richter 
                   scale rating of 8 and 5 are 108 and 105 respectively.                                         SOLUTION:  
                                 8          5                                                                    a. 
                   Divide 10  by 10 .                                                                            Substitute 220 for f  in the formula and solve for n.
                                                                                                                   
                                                                                                                  
                                                                                                                                                
                     
                   The scale rating of 8 is 103 or 1000 times greater 
                   than the scale rating of 5.                                                                   b. 
                                                                                                                 Substitute 880 for f  in the formula and solve for n.
                   b.                                                                                              
                   The amplitudes of San Francisco earthquake and                                                 
                                                                    7.8            8.1                                                          
                   New Madrid earthquake were 10                         and 10         
                   respectively. 
                     
                   Divide 108.1 by 107.8. 
                     
                    
                                                                                                                                                                             
                                                                                                                 MULTIPLE REPRESENTATIONSIn this 
                                                                                                            36.                                                                   
                                                                                                                 problem, you will explore the graphs shown:
                                                                                                                                                         
                                                                                                                 y = log4 x and 
                    
                                                                          0.3                                      
                   The New Madrid earthquake was 10                           or about 2                          
                   times greater than the San Francisco earthquake. 
                                 
                   MUSICThe first key on a piano keyboard 
              35. 
                   corresponds to a pitch with a frequency of 27.5 
                   cycles per second. With every successive key, going 
                   up the black and white keys, the pitch multiplies by a 
                   constant. The formula for the frequency of the pitch                                                                               
                   sounded when the nth note up the keyboard is played                                             
                                                                                                                  
                                                                                                                 a. ANALYTICAL
                   is given by                                                                                                                 How do the shapes of the 
                                                                                                                 graphs compare? How do the asymptotes and the x-
                   a. A note has a frequency of 220 cycles per second.                                           intercepts of the graphs compare? 
                   How many notes up the piano keyboard is this?                                                 b. VERBAL
                                                                                                                                      Describe the relationship between the 
                   b. Another pitch on the keyboard has a frequency of                                           graphs. 
                   880 cycles per second. After how many notes up the                                                
                                                                                                                 c. GRAPHICAL
                   keyboard will this be found?                                                                                              Use what you know about 
                                                                                                                 transformations of graphs to compare and contrast 
                                                                                                                                                                                              
                                                                                                                 the graph of each function and the graph of y = log4
                   SOLUTION:                                                                                         
                   a.                                                                                            x.
                                                                                                                   
                   Substitute 220 for f  in the formula and solve for n.                                          
                                                                                                                                        
                                                                                                                 1. y = log4 x + 2
                                                                                                                                             
                                                                                                                 2. y = log4 (x + 2)
                                                                                                                                      
                                                                                                                 3. y = 3 log4 x
                                                                                                                   
                                                                                                                  
                                                                                                                 d. ANALYTICAL
                                                                                                                                                Describe the relationship 
                                                                                                                                                                      
                                                                                                                 between y = log x and y =  1(log x). What are a 
                   b.                                                                                                                    4                −        4
                   Substitute 880 for f  in the formula and solve for n.                                         reasonable domain and range for each function? 
                                                                                                                     
                                                                                                                 e.ANALYTICAL
                                                                                                                                                Write an equation for a function 
                                                                                                                 for which the graph is the graph of y = log  x 
                                                                                                                                                                               3
                                                                                                                 translated 4 units left and 1 unit up. 
                                                                                                                 SOLUTION:  
                                                                                                                 a. 
                                                                                                                 The shapes of the graphs are the same. The 
                                                                                                                 asymptote for each graph is the y-axis and the x-
                                                                                                                 intercept for each graph is 1. 
                   MULTIPLE REPRESENTATIONSIn this 
              36.                                                                                                b. 
                   problem, you will explore the graphs shown:                                                   The graphs are reflections of each other over the x-
                                                           
                   y = log4 x and                                                                                axis. 
                                                                                                                     
                                                                                                                 c.
                                                                                                                 1. The second graph is the same as the first, except it
                                                                                                                 is shifted horizontally to the left 2 units. 
                                                                                                                   
                                                                                                                  
                                                        
                    
                   a. ANALYTICAL
                                                 How do the shapes of the                                                                                                 
                   graphs compare? How do the asymptotes and the x-                                                    
                                                                                                                   
                   intercepts of the graphs compare?                                                              
                                                                                                                             2, 8] scl: 1 by [ 5, 5] scl: 1 
                   b. VERBAL                                                                                             [−                       −
                                        Describe the relationship between the                                      
                   graphs.                                                                                        
                                                                                                                     
                   c. GRAPHICAL                                                                                  2. The second graph is the same as the first, except it
                                               Use what you know about                                                                                        
                   transformations of graphs to compare and contrast                                             is shifted vertically up 2 units.
                                                                                                                   
                                                                                                                  
                   the graph of each function and the graph of y = log4
                   x. 
                     
                    
                                          
                   1. y = log4 x + 2
                                               
                   2. y = log4 (x + 2)
                                        
                   3. y = 3 log4 x
                                                                                                                                                                        
                    
                                                                                                                   
                   d. ANALYTICAL                                                                                  
                                                  Describe the relationship                                                                                        
                                                                                                                            4, 8] scl: 1 by [ 5, 5] scl: 1
                                                                                                                        [−                       −
                   between y = log x and y =  1(log x). What are a 
                                           4                −        4                                             
                   reasonable domain and range for each function?                                                 
                                                                                                                 3. Each point on the second graph has a y-coordinate
                   e.ANALYTICAL
                                                  Write an equation for a function                               3 times that of the corresponding point on the first 
                   for which the graph is the graph of y = log3 x                                                graph. 
                   translated 4 units left and 1 unit up.                                                          
                                                                                                                  
                   SOLUTION:  
                   a. 
                   The shapes of the graphs are the same. The 
                   asymptote for each graph is the y-axis and the x-
                   intercept for each graph is 1. 
                   b.                                                                                                                                                  
                   The graphs are reflections of each other over the x-                                                
                   axis.                                                                                           
                                                                                                                  
                   c.                                                                                                      2, 8] scl: 1 by [ 5, 5] scl: 1 
                                                                                                                       [−                       −
                   1. The second graph is the same as the first, except it                                       d. 
                   is shifted horizontally to the left 2 units.                                                  The graphs are reflections of each other over the x-
                                                                                                                 axis. 
                    
                                                                                                                                                                            
                                                                                                                 D={x|x>0}; R={all real numbers}
                                                                                                                 e.                                   where h is the horizontal 
                                                                                                                 shift and k is the vertical shift.  Since there is a 
                                                                                                                 horizontal shift of 4 and vertical shift of 1, h = 4 and 
                                                                                                                 k = 1. 
                                                                                                                 y = log3 (x + 4) + 1 
                                                                                                                                
                                                                                                                 SOUNDThe relationship between the intensity of 
                                                                                                            37. 
                                                                                                                 sound I and the number of decibels   
                               2, 8] scl: 1 by [ 5, 5] scl: 1                                                                                                         β
                           [−                       −
                     
                                                                                                                 is                               , where I is the intensity of 
                   2. The second graph is the same as the first, except it
                   is shifted vertically up 2 units.                                                             sound in watts per square meter. 
                                                                                                                   
                                                                                                                  
                                                                                                                 a. Find the number of decibels of a sound with an 
                                                                                                                 intensity of 1 watt per square meter. 
                                                                                                                 b. Find the number of decibels of sound with an 
                                                                                                                                      −2                                      
                                                                                                                 intensity of 10          watts per square meter.
                                                                                                                     
                                                                                                                 c. The intensity of the sound of 1 watt per square 
                                                                                                                 meter is 100 times as much as the intensity of 10−2 
                                                                                                                 watts per square meter. Why are the decibels of 
                                                                                                                                                              
                                                                                                                 sound not 100 times as great?
                              4, 8] scl: 1 by [ 5, 5] scl: 1 
                          [−                       −
                                                                                                                 SOLUTION:  
                                                                                                                 a. 
                   3. Each point on the second graph has a y-coordinate                                          Substitute 1 for I in the given equation and solve for 
                   3 times that of the corresponding point on the first                                            . 
                   graph.                                                                                        β
                                                                                                                   
                                                                                                                  
                                                                                                                                                
                                                                                                                   
                                                                                                                  
                                                                                                                 b. 
                             2, 8] scl: 1 by [ 5, 5] scl: 1 
                         [−                       −                                                                                 −2
                   d.                                                                                            Substitute 10          for I in the given equation and solve 
                   The graphs are reflections of each other over the x-                                          for  . 
                   axis.                                                                                               β
                                                                                                                   
                                                                                                                  
                   D={x|x>0}; R={all real numbers}                                                                                              
                   e.                                   where h is the horizontal 
                   shift and k is the vertical shift.  Since there is a 
                   horizontal shift of 4 and vertical shift of 1, h = 4 and 
                   k = 1. 
                   y = log3 (x + 4) + 1 
                                  
                   SOUNDThe relationship between the intensity of 
              37.                                                                                                  
                   sound I and the number of decibels                                                             
                                                                        β                                            
                                                                                                                 c. Sample answer: The power of the logarithm only 
                   is                               , where I is the intensity of                                changes by 2. The power is the answer to the 
                                                                                                                 logarithm. That 2 is multiplied by the 10 before the 
                   sound in watts per square meter.                                                              logarithm. So we expect the decibels to change by 
                                                                                                                       
                                                                                                                 20.
                   a. Find the number of decibels of a sound with an 
                                                                                                                 CCSS CRITIQUE  
                   intensity of 1 watt per square meter.                                                    38.                                  Ryan and Heather are solving 
                                                                                                                 log  x    3. Is either of them correct? Explain your 
                   b. Find the number of decibels of sound with an                                                    3    ≥−
                                        −2                                                                       reasoning. 
                   intensity of 10          watts per square meter.                                                
                                                                                                                  
                   c. The intensity of the sound of 1 watt per square 
                   meter is 100 times as much as the intensity of 10−2 
                   watts per square meter. Why are the decibels of 
                   sound not 100 times as great? 
                   SOLUTION:  
                   a.                                                                                                                                               
                   Substitute 1 for I in the given equation and solve for                                         
                     . 
                   β
                     
                    
                                                  
                                                                                                                                                                    
                                                                                                                 SOLUTION:  
                                                                                                                 Sample answer: Ryan; Heather did not need to 
                                                                                                                 switch the inequality symbol when raising to a 
                                                                                                                 negative power. 
                   b. 
                                      −2                                                                                                                                                      
                                                                                                                 CHALLENGEFind log 27 +log 27 +log                                    27 +
                   Substitute 10          for I in the given equation and solve                             39.                                       3             9             27
                   for  .                                                                                                                       
                         β                                                                                       log81 27 + log243 27.
                     
                    
                                                                                                                 SOLUTION:  
                     
                    
                       
                   c.                                                                                                                    
                       Sample answer: The power of the logarithm only                                            REASONINGThe Property of Inequality for 
                                                                                                            40. 
                   changes by 2. The power is the answer to the                                                                                                                               
                   logarithm. That 2 is multiplied by the 10 before the                                          Logarithmic Functions states that when b > 1, logb x
                                                                                                                                                       
                   logarithm. So we expect the decibels to change by                                             >logby if and only if x >y. What is the case for 
                   20.                                                                                                                                                        
                                                                                                                 when 0 < b < 1? Explain your reasoning.
                                                                                                                 SOLUTION:  
                   CCSS CRITIQUE  
              38.                                  Ryan and Heather are solving 
                                                                                                                                                                                        
                   log    x       3. Is either of them correct? Explain your                                     Sample answer: When 0 < b < 1, log x > log y if 
                        3    ≥−                                                                                                                                        b           b
                   reasoning.                                                                                                          
                                                                                                                 and only if x  1, logb x                                          (s) for b in the equation y = logb 1.
                                                         
                   >logby if and only if x >y. What is the case for 
                                                                                                                 SOLUTION:  
                   when 0 < b < 1? Explain your reasoning.                                                       a. less than 
                                                                                                                 b. less than 
                   SOLUTION:                                                                                              
                                                                                                                 c. no
                   Sample answer: When 0 < b < 1, logb x > logby if                                              d. infinitely many
                                         
                   and only if x 81
                                                                                                                 SOLUTION:  
              51. log6 216                                                                                                         
                   SOLUTION:  
                   log  27 
              52.       3
                                                                                                                   4a + 6          a 
                                                                                                                 4        16
                                                                                                            57.             ≤
                   SOLUTION:  
                                                                                                                 SOLUTION:  
                                                                                                                                     
                                  
              53. 
                   SOLUTION:  
                                                                                                                                      
                                                                                                            58. 
                                                                                                                 SOLUTION:  
                   log  2401 
              54.       7
                   SOLUTION:  
                   Solve each equation or inequality. Check your 
                                  
                   solution.                                                                                                          
                     2x + 3
                   5        125                                                                             59. 
              55.             ≤
                                                                                                                 SOLUTION:  
                   SOLUTION:  
                                       
                     3x − 2           
                               
              56. 3           >81
                   SOLUTION:  
                                                                                                                                   
                                                                                                            60. 
                                                                                                                 SOLUTION:  
                     4a + 6          a 
                   4        16
              57.             ≤
                   SOLUTION:  
                                                                                                                                    
                                                                                                                 SHIPPINGThe height of a shipping cylinder is 4 
                                                                                                            61. 
                                                                                                                 feet more than the radius. If the volume of the 
                                                                                                                 cylinder is 5  cubic feet, how tall is it? Use the 
                                                                                                                                   π
                                                                                                                                       2     
                                                                                                                 formula V = πr h.
                                                                                                                 SOLUTION:  
                                                                                                                 Substitute 5  for V and r + 4 for h in the formula and
                                                                                                                                  π
              58.                                                                                                simplify. 
                   SOLUTION:                                                                                       
                                                                                                                  
                                                                                                                                                        
                                                                                                                   
                                                                                                                  
                                                                                                                 The equation has one real root r = 1. 
                                                                                                                 Thus, the height of the shipping cylinder is 1 + 4 = 5 
                                                                                                                      
                                                                                                                 ft.
                                                                                                                                                     
                                                                                                                 NUMBER THEORYTwo complex conjugate 
                                                                                                            62. 
              59.                                                                                                numbers have a sum of 12 and a product of 40. Find 
                                                                                                                 the two numbers 
                   SOLUTION:  
                                                                                                                 SOLUTION:  
                                                                                                                 The equations that represent the situation are:
                                                                                                                   
                                                                                                                  
                                                                                                                                                             
                                                                                                                   
                                                                                                                  
                                                                                                                 Solve equation (1). 
                                                                                                                   
                                                                                                                  
                                                                                                                                                   
              60. 
                   SOLUTION:  
                                                                                                                   
                                                                                                                  
                                                                                                                 Solve equation (2). 
                                                                                                                   
                                                                                                                  
                                                                                                                                                                              
                   SHIPPING The height of a shipping cylinder is 4 
              61.                                                                                                  
                                                                                                                  
                   feet more than the radius. If the volume of the                                               Thus, the two numbers are 6 + 2i and 6   2i. 
                                                                                                                                                                            –
                   cylinder is 5π cubic feet, how tall is it? Use the                                            Simplify. Assume that no variable equals zero.
                                         2     
                   formula V = πr h.                                                                               5    3 
                                                                                                            63. x · x
                   SOLUTION:  
                   Substitute 5  for V and r + 4 for h in the formula and                                        SOLUTION:  
                                    π                                                                                                 
                   simplify.
                     
                    
                                                          
                                                                                                                 a2   a6 
                                                                                                            64.       ·
                    
                   The equation has one real root r = 1.                                                         SOLUTION:  
                   Thus, the height of the shipping cylinder is 1 + 4 = 5                                                              
                        
                   ft.
                                                       
                   NUMBER THEORYTwo complex conjugate 
              62.                                                                                                (2p2n)3 
                   numbers have a sum of 12 and a product of 40. Find                                       65. 
                   the two numbers 
                                                                                                                 SOLUTION:  
                   SOLUTION:  
                   The equations that represent the situation are:
                     
                                                                                                                 (3b3c2)2 
                                                                                                            66. 
                                                                                                                 SOLUTION:  
                     
                    
                   Solve equation (1). 
                     
                                                                                                                           
                                                                                                            67. 
                                                                                                                 SOLUTION:  
                                                                                                                                     
                    
                   Solve equation (2). 
                     
                                                                                                                             
                                                                                
                                                                                                            68. 
                                                                                                                 SOLUTION:  
                                                                                                                                  
                     
                    
                   Thus, the two numbers are 6 + 2i and 6   2i. 
                                                                              –
                   Simplify. Assume that no variable equals zero.
                   x5   x3 
              63.       ·
                   SOLUTION:  
                                        
                   a2   a6 
              64.       ·
                   SOLUTION:  
                                         
                   (2p2n)3 
              65. 
                   SOLUTION:  
                   (3b3c2)2 
              66. 
                   SOLUTION:  
                             
              67. 
                   SOLUTION:  
                                       
                               
              68. 
                   SOLUTION:  
                                    
                                                                                                                 Solve each equation.
                                                                                                                                  
                                                                                                              1. 
                                                                                                                 SOLUTION:  
                                                                                                                                        
                                                                                                                                   
                                                                                                              2. 
                                                                                                                 SOLUTION:  
                                                                                                                                        
                                                                                                                                                                          2                    
                                                                                                                 MULTIPLE CHOICE
                                                                                                              3.                                       Solve log5 (x − 10) = log5
                                                                                                                       
                                                                                                                 3x.
                                                                                                                 A 10 
                                                                                                                 B   
                                                                                                                     2
                                                                                                                 C   
                                                                                                                     5
                                                                                                                 D 2, 5 
                                                                                                                 SOLUTION:  
                                                                                                                                                     
                                                                                                                   
                                                                                                                                                       
                                                                                                                   
                   Solve each equation.                                                                          Substitute each value into the original equation. 
                                                                                                                   
                1.                                                                                                                                                            
                   SOLUTION:  
                                          
                                                                                                                   
                                                                                                                 The domain of a logarithmic function cannot be 0, so 
                                                                                                                 log5 (–6) is undefined and –2 is an extraneous 
                                                                                                                 solution. 
                                                                                                                   
                                                                                                                 C is the correct option. 
                                                                                                                 Solve each inequality.
                                                                                                                              
                                                                                                              4. log5 x > 3
                2. 
                                                                                                                 SOLUTION:  
                                                                                                                                    
                   SOLUTION:  
                                          
                                                                                                                   
                                                                                                                                                          
                                                                                                                 Thus, solution set is  {x | x > 125}.
                                                                                                                                   
                                                                                                              5. log8 x ≤ −2
                                                                                                                 SOLUTION:  
                                                                            2                                                      
                   MULTIPLE CHOICE
                3.                                       Solve log5 (x − 10) = log5
                         
                   3x.
                   A 10 
                   B   
                       2                                                                                           
                   C   
                       5
                   D 2, 5                                                                                        Thus, solution set is                                 .
                   SOLUTION:  
                                                                                                                                                           
                                                                                                              6. log4 (2x + 5) ≤ log4 (4x − 3)
                                                                                                                 SOLUTION:  
                                                                                                                                                            
                     
                                                         
                                                                                                                   
                                                                                                                                                         
                                                                                                                 Thus, solution set is {x | x ≥ 4}.
                   Substitute each value into the original equation.                                                                                  
                                                                                                              7. log8 (2x) > log8 (6x − 8)
                                                                                
                                                                                                                 SOLUTION:  
                                                                                                                                                      
                     
                   The domain of a logarithmic function cannot be 0, so 
                   log5 (–6) is undefined and –2 is an extraneous 
                   solution.                                                                                       
                                                                                                                 Exclude all values of x for which 
                   C is the correct option.                                                                                                      
                                                                                                                   
                   Solve each inequality.                                                                        So,                                     
                                
                4. log5 x > 3
                                                                                                                   
                   SOLUTION:                                                                                                                                           
                                                                                                                 Thus, solution set is                               .
                                                                                                                 CCSS STRUCTURE  Solve each equation.
                                                                                                                                   
                                                                                                              8. 
                   Thus, solution set is  {x | x > 125}.
                                                                                                                 SOLUTION:  
                5. log8 x ≤ −2                                                                                                          
                   SOLUTION:  
                                     
                     
                   Thus, solution set is                                 .                                                         
                                                                                                              9. 
                                                             
                6. log4 (2x + 5) ≤ log4 (4x − 3)
                                                                                                                 SOLUTION:  
                                                                                                                                         
                   SOLUTION:  
                                                              
                     
                                                           
                   Thus, solution set is {x | x ≥ 4}.
                                                        
                7. log8 (2x) > log8 (6x − 8)                                                                                      
                                                                                                            10. 
                   SOLUTION:  
                                                                                                                 SOLUTION:  
                                                                                                                                     
                     
                   Exclude all values of x for which 
                                                   
                     
                   So,                                     
                                                                                                                                    
                                                                                                            11. 
                   Thus, solution set is                               .
                                                                                                                 SOLUTION:  
                                                                                                                                       
                   CCSS STRUCTURE  Solve each equation.
                                     
                8. 
                   SOLUTION:  
                                          
                                                                                                                                    
                                                                                                            12. 
                                                                                                                 SOLUTION:  
                                                                                                                                     
                                     
                9. 
                   SOLUTION:  
                                           
                                                                                                                                    
                                                                                                            13. 
                                                                                                                 SOLUTION:  
                                                                                                                                     
                                    
              10. 
                   SOLUTION:  
                                       
                                                                                                                                                  2        
                                                                                                            14. log3 (3x + 8) = log3 (x + x)
                                                                                                                 SOLUTION:  
                                                                                                                                                              
                                      
              11. 
                   SOLUTION:                                                                                       
                                                                                                                                                   
                                                                                                                   
                                                                                                                 Substitute each value into the original equation.
                                                                                                                   
                                                                                                                                                                              
                                      
              12.                                                                                                  
                                                                                                                 Thus, x = –2 or 4. 
                   SOLUTION:                                                                                                 2                              
                                                                                                                                                        
                                                                                                            15. log12 (x − 7) = log12 (x + 5)
                                                                                                                 SOLUTION:  
                                                                                                                                                            
                                                                                                                   
                                                                                                                  
                                                                                                                                                   
                                      
              13.                                                                                                  
                                                                                                                 Substitute each value into the original equation.
                   SOLUTION:                                                                                       
                                                                                                                                                                              
                                                                                                                   
                                                                                                                 Thus, x = –3 or 4. 
                                                                                                                           2                           
                                                                                                            16. log6 (x − 6x) = log6 (−8)
                                                                                                                 SOLUTION:  
                                                                                                                                                         
                                                    2        
              14. log3 (3x + 8) = log3 (x + x)
                   SOLUTION:  
                                                                
                                                                                                                   
                                                                                                                  
                                                                                                                                                 
                                                                                                                   
                                                                                                                 Substitute each value into the original equation. 
                                                                                                                   
                                                                                                                                                                              
                     
                   Substitute each value into the original equation.
                                                                                                                   
                                                                                                                 log6 (–8) is undefined, so 4 and 2 are extraneous 
                                                                                                                 solutions. 
                                                                                                                 Thus, no solution. 
                   Thus, x = –2 or 4.                                                                                      2                                   
                                                                                                            17. log9 (x − 4x) = log9 (3x − 10)
                               2                              
              15. log       (x −7) =log            (x + 5)
                        12                     12                                                                SOLUTION:  
                                                                                                                                                               
                   SOLUTION:  
                                                              
                                                                                                                   
                                                                                                                  
                                                                                                                                                 
                    
                                                                                                                   
                                                                                                                 Substitute each value into the original equation. 
                   Substitute each value into the original equation.                                               
                                                                                                                                                                              
                                                                                
                                                                                                                   
                                                                                                                 log9 (–4) is undefined and 2 is extraneous solution.
            7-4 Solving Logarithmic Equations and Inequalities
                                                                                                                                    
                   Thus, x = –3 or 4.                                                                            Thus, x = 5.
                             2                                                                                               2                                 
              16. log6 (x − 6x) = log6 (−8)                                                                 18. log4 (2x + 1) = log4 (10x − 7)
                   SOLUTION:                                                                                     SOLUTION:  
                                                                                                                                                               
                                                                                                                   
                                                                                                                  
                                                                                                                                                     
                                                                                                                   
                   Substitute each value into the original equation.                                             Substitute each value into the original equation.
                                                                                                                   
                                                                                                                                                                              
                                                                                                                   
                   log  (–8) is undefined, so 4 and 2 are extraneous                                             Thus, x = 1 or 4. 
                        6
                   solutions.                                                                                              2                                
                   Thus, no solution.                                                                       19. log7 (x − 4) = log7 (− x + 2)
                             2                                                                                   SOLUTION:  
              17. log9 (x − 4x) = log9 (3x − 10)                                                                                                              
                   SOLUTION:  
                                                                 
                                                                                                                   
                                                                                                                  
                                                                                                                                                     
                     
                                                                                                                   
                                                                                                                 Substitute each value into the original equation. 
                                                                                                                   
                   Substitute each value into the original equation.                                                                                                          
                     
                                                                                                                   
                                                                                                                 Since you can not have a log of 0, x =                      3 is the 
                                                                                                                 solution. 
                                                                                                                                   
                   log9 (–4) is undefined and 2 is extraneous solution.                                          SCIENCEThe equation for wind speed w, in 
                   Thus, x = 5.                                                                                  miles per hour, near the center of a tornado is w
                                                                                                                 = 93 log  d + 65, where d is the distance in 
                               2                                                                                               10
                                                                                                                                                                   
                                                                                                                 miles that the tornado travels.
              18. log4 (2x + 1) = log4 (10x − 7)                                                            20. Write this equation in exponential form.
                   SOLUTION:  
                                                                                                                 SOLUTION:  
            eSolutions Manual - Powered by Cognero                                                                                                                                      Page4
                     
                    
                                                       
                                                                                                            21. In May of 1999, a tornado devastated Oklahoma City
                   Substitute each value into the original equation.                                             with the fastest wind speed ever recorded. If the 
                                                                                                                 tornado traveled 525 miles, estimate the wind speed 
                                                                                                                 near the center of the tornado. 
                                                                                
                                                                                                                 SOLUTION:  
                                                                                                                 Substitute 525 for d in the equation and simplify.
                                                                                                                   
                   Thus, x = 1 or 4.                                                                                                             
                             2                                
              19. log7 (x − 4) = log7 (− x + 2)
                                                                                                                 Solve each inequality.
                   SOLUTION:                                                                                                       
                                                                                                            22. log6 x < −3
                                                                                                                 SOLUTION:  
                                                                                                                                     
                     
                    
                                                       
                                                                                                                 The solution set is                                  .
                     
                   Substitute each value into the original equation.                                                          
                                                                                                            23. log4 x ≥ 4
                                                                                
                                                                                                                 SOLUTION:  
                                                                                                                                     
                     
                   Since you can not have a log of 0, x =                      3 is the 
                   solution.                                                                                       
                                                                                                                                                       
                                                                                                                 The solution set is {x | x ≥ 256}.
                   SCIENCEThe equation for wind speed w, in 
                   miles per hour, near the center of a tornado is w                                                               
                   = 93 log  d + 65, where d is the distance in                                             24. log3 x ≥ −4
                                 10
                                                                     
                   miles that the tornado travels.
              20. Write this equation in exponential form.                                                       SOLUTION:  
                                                                                                                                    
                   SOLUTION:  
                                                                                                                 The solution set is                          .
                                                                                                                                   
                                                                                                            25. log2 x ≤ −2
              21. In May of 1999, a tornado devastated Oklahoma City                                             SOLUTION:  
                   with the fastest wind speed ever recorded. If the                                                                
                   tornado traveled 525 miles, estimate the wind speed 
                   near the center of the tornado. 
                   SOLUTION:  
                   Substitute 525 for d in the equation and simplify.                                            The solution set is                              .
                     
                                                   
                                                                                                                              
                                                                                                            26. log5 x > 2
                   Solve each inequality.                                                                        SOLUTION:  
                                     
              22. log6 x < −3                                                                                                      
                   SOLUTION:  
                                                                                                                   
                                                                                                                 The solution set is                         .
                                                                                                                                   
                                                                                                            27. log7 x < −1
                   The solution set is                                  .
                                                                                                                 SOLUTION:  
                                                                                                                                   
                                
              23. log4 x ≥ 4
                   SOLUTION:  
                                       
                                                                                                                 The solution set is                              .
                                                                                                                                                           
                                                                                                            28. log2 (4x − 6) > log2 (2x + 8)
                                                         
                   The solution set is {x | x ≥ 256}.
                                                                                                                 SOLUTION:  
                                                                                                                                                            
              24. log3 x ≥ −4
                   SOLUTION:  
                                      
                                                                                                                   
                                                                                                                 The solution set is                        .
                                                                                                                                                           
                   The solution set is                          .                                           29. log7 (x + 2) ≥  log7 (6x − 3)
                                                                                                                 SOLUTION:  
                                     
              25. log2 x ≤ −2                                                                                                                             
                   SOLUTION:  
                                      
                                                                                                                   
                                                                                                                 Exclude all values of x for which 
                                                                                                                                                    
                   The solution set is                              .                                              
                                                                                                                 So,                                       
                                
              26. log5 x > 2                                                                                       
                                                                                                                                                                   
                   SOLUTION:                                                                                     The solution set is                             .
                                     
                                                                                                                                                           
                                                                                                            30. log3 (7x – 6) < log3 (4x + 9)
                                                                                                                 SOLUTION:  
                   The solution set is                         .                                                                                            
                                     
              27. log7 x < −1
                   SOLUTION:                                                                                       
                                                                                                                 Exclude all values of x for which 
                                                                                                                                                      
                                                                                                                   
                                                                                                                 So,                                        
                   The solution set is                              .                                              
                                                                                                                 The solution set is                              . 
                                                             
              28. log2 (4x − 6) > log2 (2x + 8)
                   SOLUTION:                                                                                                                                 
                                                                                                            31. log5 (12x + 5) ≤ log5 (8x + 9)
                                                                                                                 SOLUTION:  
                                                                                                                                                             
                     
                   The solution set is                        .
                                                                                                                   
              29. log7 (x + 2) ≥  log7 (6x − 3)                                                                  Exclude all values of x for which 
                                                                                                                                                       
                   SOLUTION:                                                                                       
                                                                                                                                                                
                                                                                                                 So, 
                                                                                                                   
                                                                                                                 The solution set is                                  . 
                     
                   Exclude all values of x for which                                                                                                          
                                                                                                            32. log11 (3x − 24) ≥ log11 (−5x − 8)
                     
                                                                                                                 SOLUTION:  
                   So, 
                     
                   The solution set is                             . 
                                                                                                                   
              30. log     (7x – 6) < log (4x + 9)                                                                                                             
                        3                     3                                                                  The solution set is {x | x ≥ 2}.
                   SOLUTION:                                                                                                                             
                                                                                                            33. log9 (9x + 4) ≤ log9 (11x − 12)
                                                              
                                                                                                                 SOLUTION:  
                     
                   Exclude all values of x for which 
                                                                                                                   
                                                                                                                                                       
                                                                                                                 The solution set is {x | x ≥ 8}.
                   So,                                                                                                                            
                                                                                                            34. CCSS MODELINGThe magnitude of an 
                                                                                                                 earthquake is measured on a logarithmic scale called 
                                                                                                                                                                                               
                                                                                                                 the Richter scale. The magnitude M is given by M =
                   The solution set is                              .                                            log10 x, where x represents the amplitude of the 
                                                                                                                 seismic wave causing ground motion. 
                                                                                                                 a. How many times as great is the amplitude caused 
              31. log5 (12x + 5) ≤ log5 (8x + 9)                                                                 by an earthquake with a Richter scale rating of 8 as 
                                                                                                                 an aftershock with a Richter scale rating of 5? 
                   SOLUTION:                                                                                     b. In 1906, San Francisco was almost completely 
                                                                                                                 destroyed by a 7.8 magnitude earthquake. In 1911, 
                                                                                                                 an earthquake estimated at magnitude 8.1 occurred 
                                                                                                                 along the New Madrid fault in the Mississippi River 
                                                                                                                 Valley. How many times greater was the New 
                                                                                                                 Madrid earthquake than the San Francisco 
                   Exclude all values of x for which                                                             earthquake? 
                                                         
                                                                                                                 SOLUTION:  
                                                                                                                 a. 
                   So,                                                                                           The amplitude of the seismic wave with a Richter 
                                                                                                                 scale rating of 8 and 5 are 108 and 105 respectively. 
                                                                                                                 Divide 108 by 105. 
                   The solution set is                                  .                                          
                                                                                                                                   
                                                                
              32. log11 (3x − 24) ≥ log11 (−5x − 8)
                                                                                                                   
                   SOLUTION:                                                                                     The scale rating of 8 is 103 or 1000 times greater 
                                                                                                                 than the scale rating of 5. 
                                                                                                                   
                                                                                                                 b. 
                                                                                                                 The amplitudes of San Francisco earthquake and 
                                                                                                                 New Madrid earthquake were 107.8 and 108.1 
                                                                                                                 respectively. 
                   The solution set is {x | x ≥ 2}.                                                                
                                                                                                                               8.1          7.8  
              33. log9 (9x + 4) ≤ log9 (11x − 12)                                                                Divide 10          by 10       .
                                                                                                                   
                                                                                                                  
                   SOLUTION:                                                                                                            
                                                                                                                   
                                                                                                                  
                                                                                                                 The New Madrid earthquake was 100.3 or about 2 
                                                                                                                 times greater than the San Francisco earthquake. 
                                                         
                   The solution set is {x | x ≥ 8}.                                                                            
                                                                                                                 MUSICThe first key on a piano keyboard 
                                                                                                            35. 
              34. CCSS MODELINGThe magnitude of an                                                               corresponds to a pitch with a frequency of 27.5 
                   earthquake is measured on a logarithmic scale called                                          cycles per second. With every successive key, going 
                                                                                                                 up the black and white keys, the pitch multiplies by a 
                   the Richter scale. The magnitude M is given by M =                                            constant. The formula for the frequency of the pitch 
                   log10 x, where x represents the amplitude of the                                              sounded when the nth note up the keyboard is played
                   seismic wave causing ground motion.                                                                                                           
                   a. How many times as great is the amplitude caused                                            is given by 
                   by an earthquake with a Richter scale rating of 8 as                                          a. A note has a frequency of 220 cycles per second. 
                   an aftershock with a Richter scale rating of 5?                                               How many notes up the piano keyboard is this? 
                   b. In 1906, San Francisco was almost completely                                               b. Another pitch on the keyboard has a frequency of 
                   destroyed by a 7.8 magnitude earthquake. In 1911,                                             880 cycles per second. After how many notes up the
                   an earthquake estimated at magnitude 8.1 occurred                                             keyboard will this be found? 
                   along the New Madrid fault in the Mississippi River 
                   Valley. How many times greater was the New 
                   Madrid earthquake than the San Francisco                                                      SOLUTION:  
                   earthquake?                                                                                   a. 
                                                                                                                 Substitute 220 for f  in the formula and solve for n.
                                                                                                                   
                                                                                                                  
                   SOLUTION:  
                   a.                                                                                                                           
                   The amplitude of the seismic wave with a Richter 
                   scale rating of 8 and 5 are 108 and 105 respectively. 
                   Divide 108 by 105. 
                     
                                                                                                                 b. 
                                                                                                                 Substitute 880 for f  in the formula and solve for n.
                                                                                                                   
                                                                                                                  
                                                                                                                                                
                   The scale rating of 8 is 103 or 1000 times greater 
                   than the scale rating of 5. 
                     
                   b. 
                   The amplitudes of San Francisco earthquake and 
                   New Madrid earthquake were 107.8 and 108.1                                                                                                                
                                                                                                                 MULTIPLE REPRESENTATIONSIn this 
                   respectively.                                                                            36.                                                                   
                                                                                                                 problem, you will explore the graphs shown:
                                                                                                                                                         
                                 8.1          7.8                                                                y = log4 x and 
                   Divide 10          by 10       .                                                                
                                                                                                                  
                    
                                          
                     
                    
                   The New Madrid earthquake was 100.3 or about 2 
                   times greater than the San Francisco earthquake.                                                                                   
                                                                                                                   
                                                                                                                  
                   MUSICThe first key on a piano keyboard 
              35.                                                                                                a. ANALYTICAL
                   corresponds to a pitch with a frequency of 27.5                                                                             How do the shapes of the 
                   cycles per second. With every successive key, going                                           graphs compare? How do the asymptotes and the x-
                   up the black and white keys, the pitch multiplies by a                                        intercepts of the graphs compare? 
                                                                                                                 b. VERBAL
                   constant. The formula for the frequency of the pitch                                                               Describe the relationship between the 
                   sounded when the nth note up the keyboard is played                                           graphs.
                                                                                                                     
                                                                                                                 c. GRAPHICAL
                                                                                                                                             Use what you know about 
                   is given by                                                                                   transformations of graphs to compare and contrast 
                                                                                                                                                                                              
                   a. A note has a frequency of 220 cycles per second.                                           the graph of each function and the graph of y = log4
                   How many notes up the piano keyboard is this?                                                 x. 
                                                                                                                   
                   b. Another pitch on the keyboard has a frequency of                                            
                                                                                                                                        
                   880 cycles per second. After how many notes up the                                            1. y = log4 x + 2
                   keyboard will this be found?                                                                                              
                                                                                                                 2. y = log4 (x + 2)
                   SOLUTION:                                                                                                          
                                                                                                                 3. y = 3 log4 x
                   a.                                                                                              
                   Substitute 220 for f  in the formula and solve for n.                                          
                                                                                                                 d. ANALYTICAL
                                                                                                                                                Describe the relationship 
                                                                                                                                                                      
                                                                                                                 between y = log x and y =  1(log x). What are a 
                                                                                                                                         4                −        4
                                                                                                                 reasonable domain and range for each function? 
                                                                                                                     
                                                                                                                 e.ANALYTICAL
                                                                                                                                                Write an equation for a function 
                                                                                                                 for which the graph is the graph of y = log3 x 
                                                                                                                 translated 4 units left and 1 unit up. 
                   b.                                                                                            SOLUTION:  
                   Substitute 880 for f  in the formula and solve for n.                                         a. 
                                                                                                                 The shapes of the graphs are the same. The 
                                                                                                                 asymptote for each graph is the y-axis and the x-
                                                                                                                 intercept for each graph is 1. 
                                                                                                                 b. 
                                                                                                                 The graphs are reflections of each other over the x-
                                                                                                                 axis. 
                                                                                                                     
                                                                                                                 c.
                                                                                                                 1. The second graph is the same as the first, except it
                                                                                                                 is shifted horizontally to the left 2 units. 
                   MULTIPLE REPRESENTATIONSIn this 
              36.                                                                                                  
                   problem, you will explore the graphs shown:                                                    
                                                           
                   y = log4 x and 
                     
                    
                                                                                                                                                                          
                                                                                                                   
                                                                                                                  
                                                                                                                             2, 8] scl: 1 by [ 5, 5] scl: 1 
                                                                                                                         [−                       −
                                                                                                                   
                                                                                                                  
                                                                                                                 2. The second graph is the same as the first, except it
                   a. ANALYTICAL
                                                 How do the shapes of the                                        is shifted vertically up 2 units. 
                   graphs compare? How do the asymptotes and the x-                                                
                   intercepts of the graphs compare?                                                              
                   b. VERBAL
                                        Describe the relationship between the 
                   graphs.
                       
                   c. GRAPHICAL
                                               Use what you know about 
                   transformations of graphs to compare and contrast 
                                                                                                
                   the graph of each function and the graph of y = log4
                   x.                                                                                                                                                   
                                                                                                                       
                                                                                                                   
                                                                                                                  
                   1. y = log4 x + 2                                                                                        4, 8] scl: 1 by [ 5, 5] scl: 1 
                                                                                                                        [−                       −
                   2. y = log       (x + 2)                                                                        
                                 4                                                                                
                                                                                                                 3. Each point on the second graph has a y-coordinate
                   3. y = 3 log4 x                                                                               3 times that of the corresponding point on the first 
                     
                                                                                                                 graph. 
                   d. ANALYTICAL
                                                  Describe the relationship                                        
                                                                                                                  
                   between y = log x and y =  1(log x). What are a 
                                           4                −        4
                   reasonable domain and range for each function? 
                       
                   e.ANALYTICAL
                                                  Write an equation for a function 
                   for which the graph is the graph of y = log3 x 
                   translated 4 units left and 1 unit up. 
                   SOLUTION:                                                                                                                                           
                                                                                                                       
                   a.                                                                                              
                   The shapes of the graphs are the same. The                                                     
                                                                                                                           2, 8] scl: 1 by [ 5, 5] scl: 1 
                   asymptote for each graph is the y-axis and the x-                                                   [−                       −
                   intercept for each graph is 1.                                                                d. 
                   b.                                                                                            The graphs are reflections of each other over the x-
                                                                                                                 axis. 
                   The graphs are reflections of each other over the x-                                                                                                     
                   axis.                                                                                         D={x|x>0}; R={all real numbers}
                                                                                                                 e.                                   where h is the horizontal 
                   c.                                                                                            shift and k
                   1. The second graph is the same as the first, except it                                                        is the vertical shift.  Since there is a 
                   is shifted horizontally to the left 2 units.                                                  horizontal shift of 4 and vertical shift of 1, h = 4 and 
                                                                                                                 k = 1. 
                                                                                                                 y = log3 (x + 4) + 1 
                                                                                                                                
                                                                                                                 SOUNDThe relationship between the intensity of 
                                                                                                            37. 
                                                                                                                 sound I and the number of decibels   
                                                                                                                                                                      β
                                                                                                                 is                               , where I is the intensity of 
                                                                                                                 sound in watts per square meter. 
                                                                                                                   
                                                                                                                  
                                                                                                                 a. Find the number of decibels of a sound with an 
                               2, 8] scl: 1 by [ 5, 5] scl: 1 
                           [−                       −                                                            intensity of 1 watt per square meter. 
                     
                                                                                                                 b. Find the number of decibels of sound with an 
                   2. The second graph is the same as the first, except it                                                            −2                                      
                   is shifted vertically up 2 units.                                                             intensity of 10          watts per square meter.
                                                                                                                     
                                                                                                                 c. The intensity of the sound of 1 watt per square 
                                                                                                                                                                                         −2
                                                                                                                 meter is 100 times as much as the intensity of 10                           
                                                                                                                 watts per square meter. Why are the decibels of 
                                                                                                                 sound not 100 times as great? 
                                                                                                                 SOLUTION:  
                                                                                                                 a. 
                                                                                                                 Substitute 1 for I in the given equation and solve for 
                                                                                                                   . 
                                                                                                                 β
                     
                                                                                                                   
                                                                                                                  
                              4, 8] scl: 1 by [ 5, 5] scl: 1
                          [−                       −                                                                                            
                     
                    
                   3. Each point on the second graph has a y-coordinate
                   3 times that of the corresponding point on the first 
                   graph. 
                     
                    
                                                                                                                   
                                                                                                                  
                                                                                                                 b. 
                                                                                                                 Substitute 10−2 for I in the given equation and solve 
                                                                                                                 for  . 
                                                                                                                       β
                                                                                                                   
                                                                                                                  
                                                                                                                                                
                     
                    
                             2, 8] scl: 1 by [ 5, 5] scl: 1 
                         [−                       −
                   d. 
                   The graphs are reflections of each other over the x-
                   axis. 
                                                                              
                   D={x|x>0}; R={all real numbers}
                   e.                                   where h is the horizontal 
                                                                                                                   
                   shift and k                                                                                    
                                    is the vertical shift.  Since there is a 
                   horizontal shift of 4 and vertical shift of 1, h = 4 and                                          
                                                                                                                 c. Sample answer: The power of the logarithm only 
                   k = 1.                                                                                        changes by 2. The power is the answer to the 
                   y = log3 (x + 4) + 1                                                                          logarithm. That 2 is multiplied by the 10 before the 
                                                                                                                 logarithm. So we expect the decibels to change by 
                                                                                                                 20. 
                   SOUNDThe relationship between the intensity of 
              37. 
                   sound I and the number of decibels   
                                                                        β
                                                                                                                 CCSS CRITIQUE  
                                                                                                            38.                                  Ryan and Heather are solving 
                                                                                                                 log  x    3. Is either of them correct? Explain your 
                   is                               , where I is the intensity of                                     3    ≥−
                   sound in watts per square meter.                                                              reasoning. 
                                                                                                                   
                                                                                                                  
                    
                   a. Find the number of decibels of a sound with an 
                   intensity of 1 watt per square meter. 
                   b. Find the number of decibels of sound with an 
                                        −2                                      
                   intensity of 10          watts per square meter.
                       
                   c. The intensity of the sound of 1 watt per square                                                                                               
                                                                                           −2                      
                   meter is 100 times as much as the intensity of 10                                              
                   watts per square meter. Why are the decibels of 
                   sound not 100 times as great? 
                   SOLUTION:  
                   a. 
                   Substitute 1 for I in the given equation and solve for                                                                                           
                     . 
                   β
                                                                                                                 SOLUTION:  
                    
                                                                                                                 Sample answer: Ryan; Heather did not need to 
                                                                                                                 switch the inequality symbol when raising to a 
                                                                                                                 negative power. 
                                                                                                                                                                                              
                                                                                                                 CHALLENGEFind log 27 +log 27 +log                                    27 +
                                                                                                            39.                                       3             9             27
                                                                                                                                                
                                                                                                                 log81 27 + log243 27.
                                                                                                                 SOLUTION:  
                    
                   b. 
                   Substitute 10−2 for I in the given equation and solve 
                   for  . 
                         β
                     
                    
                                                  
                                                                                                                                         
                                                                                                                 REASONINGThe Property of Inequality for 
                                                                                                            40. 
                                                                                                                                                                                              
                                                                                                                 Logarithmic Functions states that when b > 1, logb x
                                                                                                                                                       
                                                                                                                 >logby if and only if x >y. What is the case for 
                                                                                                                                                                              
                                                                                                                 when 0 < b < 1? Explain your reasoning.
                    
                                                                                                                 SOLUTION:  
                   c. Sample answer: The power of the logarithm only 
                                                                                                                                                                                        
                   changes by 2. The power is the answer to the                                                  Sample answer: When 0 < b < 1, logb x > logby if 
                   logarithm. That 2 is multiplied by the 10 before the                                                                
                   logarithm. So we expect the decibels to change by                                             and only if x  1, logb x                                          d. infinitely many
                                                         
                   >logby if and only if x >y. What is the case for 
                                                                                                                                                      
                                                                                                                 WRITING IN MATHExplain why any logarithmic
                                                                                                            44. 
                   when 0 < b < 1? Explain your reasoning.                                                                                                    
                                                                                                                 function of the form y = logb x has an x-intercept of 
                   SOLUTION:                                                                                     (1, 0) and no y-intercept. 
                                                                                          
                   Sample answer: When 0 < b < 1, logb x > logby if 
                                                                                                                 SOLUTION:  
                   and only if x 81
              50. 
                                                                                                                 SOLUTION:  
                                                                                                                                   
                   SOLUTION:  
                   log  216 
              51.       6
                                                                                                                   4a + 6          a 
                                                                                                                 4        16
                   SOLUTION:                                                                                57.             ≤
                                                                                                                 SOLUTION:  
                                                                                                                                     
                   log  27 
              52.       3
                   SOLUTION:  
                                                                                                                                      
                                                                                                            58. 
              53. 
                                                                                                                 SOLUTION:  
                   SOLUTION:  
                   log  2401 
              54.       7
                   SOLUTION:                                                                                                          
                                                                                                            59. 
                                                                                                                 SOLUTION:  
                   Solve each equation or inequality. Check your 
                                  
                   solution.
                     2x + 3            
                   5        125
              55.             ≤
                   SOLUTION:  
                                       
                                                                                                                                   
                                                                                                            60. 
                                                                                                                 SOLUTION:  
                     3x − 2           
                               
              56. 3           >81
                   SOLUTION:  
                                     
                                                                                                                 SHIPPING The height of a shipping cylinder is 4 
                                                                                                            61. 
                     4a + 6          a                                                                           feet more than the radius. If the volume of the 
                   4        16                                                                                   cylinder is 5  cubic feet, how tall is it? Use the 
              57.             ≤                                                                                                    π
                                                                                                                                       2     
                                                                                                                 formula V = πr h.
                   SOLUTION:  
                                       
                                                                                                                 SOLUTION:  
                                                                                                                 Substitute 5  for V and r + 4 for h in the formula and
                                                                                                                                  π
                                                                                                                 simplify.
                                                                                                                   
                                                                                                                  
                                                                                                                                                        
                                                                                                                   
                                                                                                                  
              58.                                                                                                The equation has one real root r = 1. 
                   SOLUTION:                                                                                     Thus, the height of the shipping cylinder is 1 + 4 = 5 
                                                                                                                      
                                                                                                                 ft.
                                                                                                                                                     
                                                                                                                 NUMBER THEORYTwo complex conjugate 
                                                                                                            62. 
                                                                                                                 numbers have a sum of 12 and a product of 40. Find 
                                                                                                                 the two numbers 
                                                                                                                 SOLUTION:  
                                                                                                                 The equations that represent the situation are:
                                                                                                                   
                                                                                                                  
              59.                                                                                                                                            
                   SOLUTION:  
                                                                                                                   
                                                                                                                  
                                                                                                                 Solve equation (1). 
                                                                                                                   
                                                                                                                  
                                                                                                                                                   
                                                                                                                   
                                                                                                                  
                                                                                                                 Solve equation (2). 
                                                                                                                   
                                                                                                                  
              60.                                                                                                                                                             
                   SOLUTION:  
                                                                                                                   
                                                                                                                  
                                                                                                                 Thus, the two numbers are 6 + 2i and 6   2i. 
                                                                                                                                                                            –
                                                                                                                 Simplify. Assume that no variable equals zero.
                                                                                                                 x5   x3 
                                                                                                            63.       ·
                   SHIPPING 
              61.                     The height of a shipping cylinder is 4                                     SOLUTION:  
                   feet more than the radius. If the volume of the                                                                    
                   cylinder is 5  cubic feet, how tall is it? Use the 
                                     π
                                         2     
                   formula V = πr h.
                                                                                                                   2      6
                   SOLUTION:                                                                                     a    a  
                   Substitute 5  for V and r + 4 for h in the formula and                                   64.       ·
                                    π
                   simplify.                                                                                     SOLUTION:  
                                                                                                                                       
                    
                                                          
                                                                                                                 (2p2n)3 
                                                                                                            65. 
                    
                   The equation has one real root r = 1. 
                   Thus, the height of the shipping cylinder is 1 + 4 = 5                                        SOLUTION:  
                        
                   ft.
                                                       
                   NUMBER THEORYTwo complex conjugate 
              62.                                                                                                (3b3c2)2 
                   numbers have a sum of 12 and a product of 40. Find                                       66. 
                   the two numbers 
                                                                                                                 SOLUTION:  
                   SOLUTION:  
                   The equations that represent the situation are:
                     
                                                                                                                           
                                                                                                            67. 
                                                                                                                 SOLUTION:  
                                                                                                                                     
                   Solve equation (1). 
                     
                    
                                                                                                                             
                                                                                                            68. 
                     
                    
                   Solve equation (2).                                                                           SOLUTION:  
                                                                                                                                  
                    
                                                                                
                     
                    
                   Thus, the two numbers are 6 + 2i and 6   2i. 
                                                                              –
                   Simplify. Assume that no variable equals zero.
                   x5   x3 
              63.       ·
                   SOLUTION:  
                                        
                   a2   a6 
              64.       ·
                   SOLUTION:  
                                         
                   (2p2n)3 
              65. 
                   SOLUTION:  
                   (3b3c2)2 
              66. 
                   SOLUTION:  
                             
              67. 
                   SOLUTION:  
                                       
                               
              68. 
                   SOLUTION:  
                                    
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