NAME: ______________________________________________________ DATE: ___________________ 
                                                                          Aim: How do we use the Change of Base Rule? 
                                                                                                                   
              Learning Goals:  
              1) What is the change of base rule for logarithms? 
              2) How do we solve logarithmic equations using change of base rule? 
               
              Do Now:
              Evaluate the following:  
              a)                                                                                           b)         
                                                                                                                        
               
               
               
               
                                                                                                                   
                                                                             Change of Base Formula for Logarithms 
                       Sometimes it is necessary to rewrite logarithms with a different base.  Doing this can allow a question to 
                        be solved by using a calculator. 
                       For any positive numbers       and  , with        and        :  
                                        where c is the new base. 
               
              1)  Let’s look at the two examples from the warm-up.  Use the change of base rule to rewrite the logarithms 
                    with base 10 and then use the “LOG” key in your calculator.   
              a)                                                                                           b)         
                                                                                                                        
               
               
               
               
               
               
               
               
              2)  Let’s look at the two examples from the warm-up.  Use the change of base rule to rewrite the logarithms 
                    with base   (natural logarithm) and then use the “LN” key in your calculator.   
              a)                                                                                           b)         
                                                                                                                        
               
               
               
               
               
               
                                       Change of Base Formula 
                                                               
      3)  Use the change of base property to rewrite each logarithmic function in terms of the common logarithm 
         function (base 10). 
                                    b)    ( )       (  )          c)    ( )       ( )   
      a)    ( )       ( )                                              
                                                                               
                                                                   
                                                                   
                                                                   
                                                                   
                                                                   
                                                                   
                                                                   
       
      4)  Using the change of base rule, rewrite the given logarithmic expressions in base 7.  
      a)                            b)                            c)          (    )    
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                   
                                                                   
                                                                   
       
      5)  Rewrite each logarithmic function as a natural logarithm function (base  ). 
      a)   ( )       (  )                                                       
                                    b)   ( )      (    )         c)   ( )        ( ) 
                                                                                
                                                                  
                                                                  
                                                                  
                                                                  
                                                                  
                                                                  
                                                                  
                                                      Change of Base Formula 
                                                                                        
                          USING THE CHANGE OF BASE FORMULA TO SOLVE LOG EQUATIONS 
            What is the difference between the two questions below? 
                          (     )       (     )                                          (     )       (     ) 
                                                                        
                                                                        
                                                                        
                                                                        
                                                                        
                                                                        
          
            How can we use the change of base formula to solve a logarithmic equation with different bases?  
             1)  Use the change of base formula to rewrite the logarithmic expression to all have the same base. 
                       Usually the larger base written in terms of the smaller base. 
             2)  Look to reduce the denominator after you have used the change of base formula. 
             3)  Cross multiply and simplify using the properties of logarithms 
             4)  Solve for x and check 
                  
                             ( )                
         1)  Solve for x:                   (       ) 
                                           
          
          
          
          
          
          
          
          
          
          
          
                                (       )            
          2)  Solve for x:                (       ) 
                                                 
          
          
          
          
          
          
          
          
          
                              (              )
         3)  Solve for x:                   (   ) 
                                                      
                                                                                    
         
         
         
          
              
         5)  Algebraically solve for x: