Logarithms

1. The Definition of the Logarithm
   
   

   Definition  The function logbx is defined as the inverse function of y = bx

   Recall that by definition, if f and g are inverse functions then 
   
           f(g(x)) = g(f(x)) = x

   Hence we have the following two properties:
   
   

   Log Properties  (From the inverse definition) logbbx  =  x blogb(x) = x

   
   Example:  
   
   Solve 
   
           2^x = 128

   Solution  
   
   Take the log base 2 of both sides:

           log₂2^x = log₂128 
   
   hence

           x = log₂128 
   
   Note that Property 1 allows us to cancel the log and the exponent

   
   

   Example:  
   
           log₃9 = 2 
   
   since 
   
           3² = 9

    

   Exercises:  
   
   Find
   

   1. log₁₀1000
      

   2. log₄64
      

   3. log₅1/5
      

   4. log₃( )

   Simplify
   

   1. 10^log10(1/x) 
      

   2. log₃ 27^x-1
      

   3. log₄(2^4x-2)  
      
        
      

2. Logs and Calculators
   

   Goal:  
   
   Find 
   
           log₃17
   

   Note:  The calculator has ln and log
   
   
   

   Definition log x = log10 x  ln x = loge x

    

   Change of Base Formula logba = lna/lnb = loga/logb

   Hence   
   
           log₃17 = ln17/ln3 = 2.5789...
   
   

   Exercise:  
   
   Find 
   
           log₅29 
   
   and 
   
           log₆18
   
   
   

3.  Logs and Graphs
   
   
   Below is the graph of 
   
           y = log₂ x 
   
   It can be found by reflecting 
   
            y  =  2^x  
   
   across the line 
   
           y  =  x
   
   

   

   
   
   Note:  The domain of the inverse is the range of the function and the range of the inverse is the domain of the function.  Hence, 
   
   the domain of log x is (0 , ) 
   
   and 
   
   the range of log x is R    
   
   

    

   Exercise

   Use shifting rules to graph

           y  =  log₂(x - 3) + 1 
   
   and 
   
           y  =  -log₂x  
   
   
   

4. Application
   
   The pH of a liquid describes how acidic or basic the liquid is.  Chemists define the pH by the formula:

   pH = -log[H⁺]

   where  [H⁺] is the concentration of hydrogen ions.

    

   Example
   

   A solution of Hydrochloric acid has
   
           [H⁺] = 3.2 X 10^-4
   
   Find the pH of the solution.
   

   Solution

   
           PH = -log(3.2 X 10^-4)  =  3.5

   
   
   Exercise
   
   Suppose that the pH of a shampoo is 7.3.  Find the concentration of hydrogen ions.  

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