Solving Exponential Equations • An exponential equation is an equation with a variable as part of an exponent. The following examples will show how to solve this type of equation.

• Example 1:

3

x  • Method A Take a log base 3 of each side.

Use the inverse property.

log

3

3

x 

x



log log

3 3

7 7

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Solving Exponential Equations

log

3

7

(nearest hundredth), use the change of base formula on the calculator.

log 7

3 

ln 7 ln 3

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Solving Exponential Equations • Method B Original problem.

Take the natural log of each side.

Use the Power property.

Divide each side by ln 3.

ln 3

x

3

x  

7 ln 7

x ln 3  ln 7

x



ln 7 ln 3

• Note that the result is the same as that found using method A.

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Solving Exponential Equations • Example 2: Solve the following equation.

6



5

x  2 

9



21

Get the exponential expression on the left side by itself.

6



5

x  2 

12 5

x  2 

2

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Solving Exponential Equations Now solve using one of the two methods.

Method A 5

x

 2  2 log 5 5

x

 2  log 2 5

x x

log 2 5   5 Method B Table of Contents 5

x

 2  2  ln 5

x

 2

x

  ln 2  ln 2

x x

ln 2 ln 5 ln 2 ln 5 Slide 5

Solving Exponential Equations Method B is the best way to solve an exponential equation that has a non-integer base. Consider the following equation:

1.327

x

 3 

12

Solve using Method B

1.327

x

 3 

12



ln1.327

x

 3

x

 

ln12



ln 2

x x

ln 2 ln1.327

ln 2 ln1.327

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Solving Exponential Equations Table of Contents