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File: Solving Equations Pdf 175273 | 4 Explog Solve
section 4 solving exponential and logarithmic equations solving equations with the same base to solve an exponential equation the following property is sometimes helpful if a 0 a 1 and ...

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                     Section 4:  Solving Exponential and Logarithmic 
                                             Equations 
                 
                 
                SOLVING EQUATIONS WITH THE SAME BASE 
                To solve an exponential equation, the following property is sometimes helpful: 
                                  If a > 0, a ≠1, and ax = ay , then  x = y .   
                 
                Similarly, we have the following property for logarithms: 
                                      If  log x = log y , then  x = y . 
                 
                _______________________________________________________________________ 
                Example:  Solve )log (5x−6) = log (x+2  for x: 
                                 3           3
                Solution: 
                                          log3(5x−6)=log3(x+2)
                                          5x−6=x+2
                                          5x−x=2+6              
                                          4x=8
                                          x = 2
                _______________________________________________________________________ 
                _______________________________________________________________________ 
                Example:   Solve 2x+1 =8for x. 
                Solution:  Here, the bases are not the same, but we find that we are able to manipulate the 
                right hand side to make the bases the same. 
                                                 2x+1 =8
                                                 2x+1 = 23  
                                                 x+1=3
                                                 x = 2
                _______________________________________________________________________ 
                 
                 
                 
                CANCELLATION LAWS 
                In the “Introduction to Exponential and Logarithmic Functions” file, we learned that logs 
                and exponentials are inverse functions. Since the exponential and logarithmic functions 
                are inverse functions, cancellation laws apply to give: 
                                      log (ax)= x for all real numbers x 
                                         a
                                          aloga x = x  for all  x > 0  
                 
                  We already stated that e is the most convenient base to work with for exponential and 
                  logarithmic functions.  The same cancellation laws apply for the natural exponential and 
                  the natural logarithm: 
                                           ln(ex) = x for all real numbers x 
                                                elnx = x for allx >0  
                   
                  These last two cancellation laws will be especially useful in calculus this year. To solve a 
                  simple exponential equation, you can take the natural logarithm of both sides. 
                  (technically, you can take the logarithm with any base, but the natural log is often the 
                  easiest).   Similarly, to solve a simple logarithmic equation, you can take the natural 
                  exponential of both sides.  At this point, the equation can be solved using basic algebra. 
                          
                  _______________________________________________________________________ 
                  Example:  Solve  e2x = 8 for x. 
                  Solution: 
                                                    e2x = 8
                                                    ln(e2x) = ln(8)
                                                    2x = ln(8)      
                                                     x = ln(8)
                                                          2
                  _______________________________________________________________________ 
                  _______________________________________________________________________ 
                  Example:  Solve  ln(x + 5) = 4 for x. 
                  Solution: 
                                                     ln(x + 5) = 4
                                                     eln(x+5) = e4
                                                     x + 5 = e4    
                                                     x = e4 −5
                  _______________________________________________________________________ 
                   
                   
                          
                  For a detailed explanation of some more difficult examples, check out the mini-clips!  
                   
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...Section solving exponential and logarithmic equations with the same base to solve an equation following property is sometimes helpful if a ax ay then x y similarly we have for logarithms log example solution here bases are not but find that able manipulate right hand side make cancellation laws in introduction functions file learned logs exponentials inverse since apply give all real numbers aloga already stated e most convenient work natural logarithm ln ex elnx allx these last two will be especially useful calculus this year simple you can take of both sides technically any often easiest at point solved using basic algebra eln detailed explanation some more difficult examples check out mini clips...

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