4.6 (Part A) Exponential and Logarithmic Equations  
                      
             In this section you will learn to: 
                     solve exponential equations using like bases 
                     solve exponential equations using logarithms 
                     solve logarithmic equations using the definition of a logarithm 
                     solve logarithmic equations using 1-to-1 properties of logarithms 
                     apply logarithmic and exponential equations to real-world problems 
                                       x
                     convert y = ab   to an exponential equation using base e 
              
                                                                    
                                                                                                                            y
                    Definition of a Logarithm 
                                                                              y  log  x      is equivalent to       b         x 
                                                                                     b
                                        
                                                                    
                                                                                        x                              log x
                                                                                                                          b
                          Inverse Properties 
                                                                               log b  x                             b         x
                                                                                                                                    
                                                                                    b
                                        
                                                                    
                 Log Properties Involving One 
                                                                               log b 1                               log 1 0
                                                                                                                                    
                                                                                    b                                     b
                                        
                                                                    
                             Product Rule                                            log (MN)  log M log N
                                                                                                                              
                                                                                         b                b            b
                                        
                                                                    
                                                                    
                             Quotient Rule 
                                                                                             M
                                                                                                
                                                                                     log           log M log N
                                                                                                
                                                                                                                              
                                                                                         b                b            b
                                        
                                                                                             N
                                                                                                
                                                                                                  
                                                                                                     p
                               Power Rule 
                                                                                           log M  plog M
                                                                                                                         
                                                                                               b                  b
                                        
                                                                                                       
                                                                                          M       N
                                        
                                                                                     If b     b  then M  N. 
                                        
                                                                                                       
                                        
                                                                                                       
                      One-to-One Properties 
                                                                              If log M  log N  then M  N.  
                                                                                       b            b
                                                                                                       
                                                                                                       
                                                                               If  M  N  then log M  log N . 
                                                                                                          b            b
                                                                                                       
              
              
              
              
              
              
              
                                                                Page 1 (Section 4.6) 
             Example 1:  Solve each equation by expressing each side as a power of the same base. 
                                                                                                                                 6
                                                                                  1                                             e
                       x1      x3                                        2x                                           2  x
                (a)  5    25                                       (b)  9                                       (c)  e e          
                                                                                                                                  x
                                                                                 5
                                                                                                                                e
                                                                                   3
              
              
              
              
              
              
              
              
              
                                                                                                                  2x
             Steps for solving EXPONENTIAL EQUATIONS:                                 Example 2:  Solve 5e           60 
             (Examples 2 – 6) 
              
             1.  Isolate the exponential “factor”.                                                                          
              
             2.  Take the common/natural log of both sides. 
              
                                             x                 x
             3.  Simplify  (Recall: lnb  xlnb;            lne  x) 
              
             4.  Solve for the variable. 
              
             5.  Check your answer. 
              
              
              
              
              
              
              
                                        x
             Example 3:   Solve 3  30using (a) common logarithms, (b) natural logarithms, and (c) the definition of 
             a logarithm. 
              
              
              
              
              
              
              
              
              
              
              
              
              
              
              
              
                                                                  Page 2 (Section 4.6) 
                                               x                                                                              x2
               Example 4:    Solve 10 3835                                                     Example 5:  Solve 5               50  
                
                
                
                
                
                
                
                
                
                
                
                
                
                                            x2       x1
               Example 6:  Solve 2               3                                               
                
                
                
                
                
                
                
                
                
                
                
                
                
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               Example 7:   Use FACTORING to solve each of the following equations.  (Hint:  Use substitution or 
               short-cut method learned in Section 1.6.) 
                
                         2x        x                                                                    2x         x
                  (a)   e   2e 30                                                             (b) 3     43 120                                 
                
                
                
                
                
                
                
                
                
                                                                   
                
                
                                               
                
                
                
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                                                                          Page 3 (Section 4.6) 
          Steps for solving LOGARITHMIC EQUATIONS:              Example 8:   Solve log (x3)  2 
                                                                                      4
          (Examples 8 – 11) 
           
          1.  Write as a single logarithm.  (log M  c) 
                                           b
           
                                          c
          2.  Change to exponential form.  (b  M ) 
           
          3.  Solve for the variable.   
           
          4.  Check your answer.                          
           
           
                                             
          Example 9:   Solve  log xlog (x7) 3 
                                2       2
           
           
           
           
           
           
           
           
           
           
          Example 10:  Solve 3ln2x 12                       Example 11:  Solve log (x  2)log (x 5) 1 
                                                                                  2           2
           
           
           
           
           
           
           
           
           
           
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          Steps for solving equations using 1-to-1 properties:     Example 12: log(x 7)log3  log(7x1) 
          (Examples 12 – 14) 
           
          1.  Write the equation in log M  log N  form. 
                                    b        b
           
          2.  Use 1-to-1 property.  (Write without logarithms.) 
           
          3.   Solve for the variable. 
           
          4.  Check your answer. 
                                                          
           
                                                 Page 4 (Section 4.6)