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 . 
                 
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                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
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                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
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                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. 
                          
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                  Example:  Solve  e2x = 8 for x. 
                  Solution: 
                                                    e2x = 8
                                                    ln(e2x) = ln(8)
                                                    2x = ln(8)      
                                                     x = ln(8)
                                                          2
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                  Example:  Solve  ln(x + 5) = 4 for x. 
                  Solution: 
                                                     ln(x + 5) = 4
                                                     eln(x+5) = e4
                                                     x + 5 = e4    
                                                     x = e4 −5
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                  For a detailed explanation of some more difficult examples, check out the mini-clips!