TABLE OF CONTENTS

3.2 Logarithms with Bases other than 10 and Basic Properties of Logs

Goals:

·         Find logarithms to bases other than 10

·         Develop the  basic properties of logarithms

Prerequisite skills and knowledge:

·         a working knowledge of logarithms base 10

·         a working knowledge of basic properties of exponents

Terms to know:

• Base (of a common logarithm)

• Common logarithm

• Exponential form

We can consider logarithms with bases other than 10.  

Example 1.   Discuss (guess) in small groups the meaning of the following, then evaluate:

a)                                           b)                             

c)                                        d)                                                  

e)                                           f)                                           

SOLUTION

            a)          

            b)        

The notation ,means the power you raise a to in order to get x

Example 2.  Write each of the following in exponential form:

a)    Since the base is 5 and the exponent is 2,               

      we have:                                                                           

b)   Since the base is 2 and the exponent is 5,               

      we have:                                                                           

c)    Since the base is 3 and the exponent is -3,              

      we have:                                                                           

d)   Since the base is 10 and the exponent is 2,             

      we have:                                                                           

Example 3.  Let’s go in the reverse direction.:  Write each of the following as a logarithmic equation:

SOLUTION

a)  Since the base is 5 and the exponent is 2, we have:                        

b)  Since the base is 2 and he exponent is 5, we have:                          

c)  Since the base is 3 and the exponent is -3, we have:                        

d) Since the base is 10 and the exponent is 3, we have:                         

Checkpoint Logarithms                                       

You may have discovered some interesting patterns when working on the Checkpoint exercises.

Example 4.  Look at problems 14 -17 from the Checkpoint exercise.  Can you generalize your results?              

14)                                                        15)   

16)                                                       17)   

    SOLUTION

     14)  

            Since a logarithm is an exponent and the base is 10,

           we want the power to which we take 10 that will give us 1:                                       

            Since any number to the 0 power is 1, the logarithm is 0:                                           

       15)  

            Since a logarithm is an exponent and the base is 4,

           we want the power to which we take 4 that will give us 1:                               

            Since any number to the 0 power is 1, the logarithm is 0:         

      16)   

           Since a logarithm is an exponent and the base is 3,

           we want the power to which we take 3 that will give us 1:                               

            Since any number to the 0 power is 1, the logarithm is 0:         

      17)  

            Since a logarithm is an exponent and the base is 5,

           we want the power to which we take 5 that will give us 1:                               

            Since any number to the 0 power is 1, the logarithm is 0:         

Now look at the following problems and note any pattern.

Example 5.    

SOLUTION.

Now try these.

Example 6. 

SOLUTION.

a)   

       Since the base is 5, we want the exponent to which we must take 5 in order to get 5² . 

       This is almost a silly question because the answer is really right in the problem:

b)   

      Since the base is 4, we want the exponent to which we must take 4 in order to get 4³ .       

 c)   

     Since the base is 3, we want the exponent to which we must take 3 in order to get 3².

More worked examples

Homework problems

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