6.6 Solving Logarithmic Equations 
                                        Core Concept:  If log   x = log  y, then x = y.  (In plain English, if the log bases are the same, 
                                                                                                 b                      b 
                                        then the arguments are equal to each other.) 
                                        **hint:  remember that log x means log   x and ln x means log   x 
                                                                                                                                   10                                                  e
                                                                                                                       
                                        1) Example:  log  x = log   15 Since the bases are the same, you can set the arguments equal to 
                                                                                     6                     6
                                                                                                                        each other.   
                                                                                so…x = 15 
                                                      a)                                                                b)                                                                                                 c)                                                               
                                        Try:               log  x = log   24                                                 log (7x – 4) = log (2x + 11)                                                                      ln (5x + 2) = ln (x + 10) 
                                                                     2                     2
                                         
                                         
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                                        Sometimes there is a log on only one side.  We need a way to eliminate that log.  Recall that 
                                        logs and exponents are inverses of each other, so they cancel each other out. 
                                        2) Example:  log   x = 2                                                        Since the base of the log is 8, we can undo (or cancel) the log by 
                                                                                     8
                                                                                                                        exponentiating it, or raising it to a power, with a matching base of 
                                                                                                                        8.  Do that on both sides. 
                                                                     log        x          2
                                                                  8              = 8                                    Notice the bases are now the same.  The base 8 on the left cancels 
                                                                            8
                                                                                                                        the log base 8, and the x drops to the ground. 
                                                          so…x = 64 
                                         
                                        3) Example:  log (8x – 9) = 4  Since the base of the log is 10, we will raise it to the exponent slot 
                                                                                                                        and make the base a 10. 
                                                                          log (8x – 9)                     4
                                                          10                                   = 10                     Notice the base on both sides is 10, with the original problem up 
                                                                                                                        in the exponents.  Cancel the base 10 and log base 10 on the left.  
                                                                                                                        Simplify the right. 
                                                                                                                
                                                            8x – 9 = 10,000                                             Solve for x. 
                                                                                             ଵ଴,଴଴ଽ
                                                                     x =                                    
                                                                                                   ଼
                                                                
                                                      d)                                                                                    e)                                                                                                 f)  
                                        Try:                log   x = 7                                                                          log   (x – 6) = 5                                                                                  log (2x + 1) = 2 
                                                                     5                                                                                     2
                                         
                        Sometimes there is more than one log on one side.  You will condense them to a single log, 
                        then solve as in the previous examples. 
                        4) Example:  log   x + log   (x + 12)  = 3                              We need to condense the left side to a single log.  
                                                    4            4
                                                                                                Recall log m + log n = log mn (product property). 
                                                log   x(x + 12) = 3                             Now exponentiate both sides to cancel the log. 
                                                      4
                                                   log   x(x + 12)      3
                                                4                  = 4                          Cancel the base and the log on the left.  Drop the 
                                                       4
                                                                                                argument to the ground. 
                                                x(x + 12) = 64                                  Distribute the x, then solve the quadratic equation 
                                                                                                using any of our quadratic equation solving 
                                                                                                methods.  (Remember the orange slip?) 
                                                  2
                                                x  + 12x – 64 = 0 
                                                (x – 4)(x + 16) = 0 
                                                x = 4 and x = -16                               You must check for extraneous solutions.  Notice 
                                                                                                that when you substitute -16 into the original 
                                                                                                equation, the argument of the logs is a negative 
                                                                                                number. That means x = -16 is extraneous. 
                                                so… x = 4 
                        Try: *hint…some require the quadratic formula   
                        g)                                              h)                                                          i)  
                            log   x + log   (x – 5)  = 2                    log   (x + 4) + log   (x + 1)  = 2                         ln x + ln (x – 2)  = 5 
                                  6            6                                  5                     5
                         
                         
                         
                         
                         
                         
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                        Answers: 
                              a)    x = 24                  b)  x = 3                           c)  x = 2 
                                                                                                         ଽଽ
                              d)  x = 78,125                e)  x = 38                          f)  x =     
                                                                                                         ଶ
                              g)  x = 9                     h)  x ≈ 2.720                       i)  x ≈13.2