TABLE OF CONTENTS

3.5   Solving Logarithmic Equations

Goals:   To solve logarithmic equations of the form:  where .  

Terms to know:

• Base (of exponent)

• Exponent

• Exponential form

• Logarithm
• Logarithmic equation

Skill Prep: Exponential form vs. logarithmic form

The simplest type of logarithmic equation is one like 

                                                                      .

To solve logarithmic equations you need to understand the basic relationship between exponential and logarithmic equations.  You’ve been working with them for a while, now.  Let’s review this basic relationship.

                                                                            exponent

                                                                          base

Example 1. Change the following to logarithmic form:

SOLUTION

a)   

            The base is 5, the exponent is 2.  Since the logarithm

            is the exponent, we can write it like this:                                                 

b)   

            The base is 3, the exponent is 4.  Since the logarithmic

            is the exponent, we can write it like this:                                                 

c)   

            The base is 10, the exponent is 3.  Since the logarithmic

            is the exponent, we can write it like this:                                                 

d)   

            The base is 2, the exponent is -3.  Since the logarithm           

            is the exponent, we can write it like this:                                                 

Example 2. Change the following to exponential form:

SOLUTION

a)   The base is 4, the exponent (logarithm) is 2, so                  

b)  The base is 3, the exponent (logarithm) is -1, so                  

c)  The base is 10, the exponent (logarithm) is 2, so                  

d)  The base is 5, the exponent (logarithm) is , so                   

Recall that the log of a negative number is undefined.  For example, log(-100).  Also the base needs to be a positive number, but not = 1.

Example 3.  Solve the following equations

a)   

            We use the definition of logarithm here as we

            convert to exponential form.                                                           

           Check:                                                                                                                                                   

          Graphical check

b)   

            We use the definition of logarithm and convert to

            exponential form:                                                                            

           This now is an exponential equation:  write both

           sides with the same base:                                                          

            Check:                                                                                                           

            Graphical check

c)   

            We use the definition of logarithm and convert to

            exponential form:                                                                              

            Solve the quadratic as usual:                                                                                                                                                                                                            

            Check:                                                                               

Graphical check

Checkpoint Log Equations

More worked examples

Homework problems

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