3.3 Notes Honors Precalculus                                                     Date: __________________________________ 
           
          I.  Properties of Logs 
           
                                        
          Relationship between 
          Logarithmic Function & 
          Exponential Function 
           
           
          Ex. 1    Without a calculator, put the following log expressions in order from least to greatest. 
           
                                                        log 10, log 10, log 24, log ⎛1⎞  
                                                            2          3         5          2 ⎜  ⎟
                                                                                              ⎝ 3⎠
           
           
           
           
          II. Properties of Logs 
           
                                                Logarithm with base b                                       Natural Logarithm 
          Product Property        log    uv = 
                                      b (   )                                              ln(uv) =  
                                                                                             
          Quotient Property            ⎛ u⎞                                                   u
                                  logb⎜ ⎟ =                                                ln    = 
                                       ⎝ v⎠                                                   v
          Power Property 
                                          n
                                  logb(u )=                                                lnun = 
                                                                                             
           
          Proof: 
           
           
           
           
           
           
           
           
           
           
           
           
           
           
           
           
           
           
           
           
           
         Ex. 2    Simplify the following log expressions: 
          
               ⎛ 6 ⎞                                                                      2   4
         a)  ln⎜    ⎟                                                         b)  log (4 ⋅3 ) 
               ⎝ e2 ⎠                                                                2
          
          
          
          
         III.  Change of base 
          
         Ex. 3    5x = 7 
          
          
          
          
          
          
          
          
          
          
          
          
          
          
          
                                        
          
         Change of base formula: 
          
          
          
         Ex. 4    Solve the following (approximate answers to two decimal places) 
          
         a)  log 9                                                   b)  log   3 
                 2                                                           1.5
          
          
          
          
          
          
          
          
         c)   log7 8                                                 d)  log 3 30  
          
          
          
          
          
          
          
         Ex. 5    Find exact values without a calculator. 
          
         a)  log ⎛ 1 ⎞                                               b)  log 8 − log 2 
                 5 ⎜    ⎟                                                    4        4
                  ⎝125⎠
          
          
          
          
          
         c)   2lne6 − lne5                                                    d)  log 2 + log 5 − log 40 
                                                                                     4         4         4
          
          
          
          
          
         Ex. 6    Expand the following log expressions: 
          
                  ⎛ yz3⎞                                                            ⎛   3        ⎞
         a)  log                                                              b)  ln  4   (y−1)  
                 3 ⎜ 2x ⎟                                                           ⎜   x        ⎟
                  ⎝     ⎠                                                           ⎝            ⎠
          
          
          
          
          
          
          
          
          
          
          
          
         Ex. 7    Condense the following log expressions: 
          
         a)   2                                                               b)                                      
               (ln(x +1)− ln4)                                                    log2 x + 4log2 y − 3log2(z + 4)
             5
          
          
          
          
          
          
          
          
          
          
          
          
          
          
          
         Ex. 8    Let logb 2 = A and logb 6 = B.  Express each of the following in terms of A and/or B. 
          
         a)  logb 8                                                           b)  logb12  
          
          
          
          
          
          
          
          
                                                                                       ⎛ b ⎞
         c)   logb 3                                                          d)  logb ⎜    ⎟  
                                                                                       ⎝ 36⎠