Logarithmic Functions

Definition
For  ,   is equivalent to   .

is read "the logarithm of to the base ".
Note that the values of the logarithm are exponents.
The logarithm of to the base is the power
to which must be raised in order to get .

Example
 because   .

The logarithmic function and the exponential functions are inverse functions of each other.


                                
                                                            

For     and   ,
the domain of the function is the set of all positive real number,
the range of the function is the set of all real numbers, positive,
negative, and zero.

Look at the graph of .
It is typical.

See Examples 1 - 3, pages 378 - 379.


Properties of the Logarithm
For   ,  ,    positive real numbers, and   ,    real numbers,
1)
     
                (The log of a product is equal to the sum of the logs.)
2)
     
                (The log of a quotient is equal to the difference of the logs.)

3)
     
                 (The log of a power is equal to the power times the log.)

4)
         if and only if 
5)
      
6)
         
7)
        
8)
        

Let us prove Property 1:
Let    and    .
By definition, these are equivalent to    and   .

     
                          

The other properties are proven in the same way.

Property 4 is true because the logarithmic function is one-to-one.

See Examples 4 - 7, pages 381 - 383.


CAUTION
                                  
                      
                           

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