TABLE OF CONTENTS

2.5  Solving Exponential Equations

Goal:  The student will be able to

• solve exponential equations of the type .  

Terms to know:  

·         base of the exponential

·         exponential equation

The simplest type of exponential equation is one like 

                                                                   .                                                                  

We can write the right hand side as a power of two, like this:

Since both sides have the same base, we can simply set the exponents equal to each other, i.e.

We can check algebraically;

To solve graphically, we look for the point of intersection of the graphs of each

side of the equation. We let  and .

The two graphs intersect at the point with .  When x = 3, ;  and y =  = 8.   So  is our solution. 

We can also check by looking at a table of values listing the powers of two. 

Example 1. Solve each of the following exponential equations:

a)                                  b)                               c)   

SOLUTIONS.

a)          

            Note that .  Since the 9 is in the denominator,                 

            we use a negative exponent. 

           Rewrite both sides with the same base:                                           

            Set the exponents equal to each other.                                           

            Check:                                                                                                   

Graphical Solution

Check with Table of Values

Graphical Solution

c)    

       Note that both 4 and 8 can be expressed as powers of two.

       To use our method, bring one term to the right:                                          

       Express both sides using the same base, which is 2:                         

       Use the properties of exponents to simplify:

       Power to a power  MULTIPLY exponents                                                              

      Since the bases are the same, set the exponents equal to

      each other.                                                                                                              

      Solve:                                                                                                                                                

      Check:                                                                                          

Example 2.   By using the substitution , find the value(s) of x such that  .

SOLUTION

We want to use the substitution , but we have a term with a   in it.  We need to deal with that first.                                                     

Note by the rules of exponents,   , so we can write the original equation as

We know from the definition of negative exponents that

, so we can now write:                                                            

NOW, we can rewrite the equation using the substitution :                                                                                                                                                                       

                                                     Solve for u:                                                                                                                                                 

Now, however, we need to switch back to find x.

We use the original substitution                     

The check is left for you.

           Checkpoint Exponential Equations

 More worked examples

 Homework problems

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