Table of Contents
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
Table of Contents
Solving Exponential Equations
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