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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:

Solve the equation .73x • Method A

Take a log base 3 of each side. 7log3log 3x

3 Use the inverse property. 7logx 3

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Solving Exponential Equations

The solution is . To find the answer as a decimal (nearest hundredth), use the change of base formula on the calculator.

7log3

The answer is . 77.1x

3

ln 7log 7

ln3

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Solving Exponential Equations

• Method B

Original problem. 73x Take the natural log of each side. 7ln3ln x Use the Power property. 7ln3lnx

Divide each side by ln 3.3ln

7lnx

• 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. 21956 2x

Get the exponential expression on the left side by itself.

1256 2x

25 2x

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Solving Exponential Equations

Now solve using one of the two methods.

Method A Method B2

25 5

5

5

5 2

log 5 log 2

2 log 2

2 log 2

x

x

x

x

2

2

5 2

ln5 ln 2

2 ln5 ln 2

ln 22

ln5ln 2

2ln5

x

x

x

x

x

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Solving Exponential Equations

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

31.327 12x

Solve using Method B

3

3

1.327 12

ln1.327 ln12

3 ln1.327 ln 2

x

x

x

ln 23

ln1.327ln 2

3ln1.327

x

x

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Solving Exponential Equations