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Solving Exponential Equations That Contain Like Bases

Author: Sophia
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what's covered
In this lesson, you will solve exponential equations in which the bases are equal, or can be written in such a way that they are equal. Specifically, this lesson will cover:

Table of Contents

1. Solving Exponential Equations With Like Bases

By the one-to-one property of exponential functions, an equation of the form b to the power of R equals b to the power of S can be solved by setting R equals S.

EXAMPLE

Solve the equation e to the power of 2 x minus 1 end exponent equals e to the power of 3 minus 5 x end exponent.

e to the power of 2 x minus 1 end exponent equals e to the power of 3 minus 5 x end exponent This is the original equation.
2 x minus 1 equals 3 minus 5 x Apply the one-to-one property.
7 x minus 1 equals 3
7 x equals 4
x equals 4 over 7 		Solve for x.

The solution is x equals 4 over 7.

try it
Consider the equation 3 to the power of 1 minus x end exponent equals 3 to the power of 4 x plus 8 end exponent.
Solve the equation for x.
Since the bases are equal, set the exponents equal to each other, then solve:

1 minus x equals 4 x plus 8 Setting exponents equal to each other.
1 equals 5 x plus 8 Add x to both sides.
short dash 7 equals 5 x Subtract 8 from both sides.
short dash 7 over 5 equals x Divide both sides by 5. This is the solution.

The solution to the equation is x equals short dash 7 over 5.

2. Solving Exponential Equations by Changing One or Both Bases

Sometimes the bases are not equal, but it is possible for the exponential functions to be written so that they have the same base.

EXAMPLE

Solve the equation 2 to the power of x minus 3 end exponent equals 4 to the power of 3 x minus 2 end exponent.

Note that the bases are not the same, but 4 is a power of 2.

2 to the power of x minus 3 end exponent equals 4 to the power of 3 x minus 2 end exponent This is the original equation.
2 to the power of x minus 3 end exponent equals open parentheses 2 squared close parentheses to the power of 3 x minus 2 end exponent Rewrite 4 equals 2 squared so that the bases on both sides are equal.
2 to the power of x minus 3 end exponent equals 2 to the power of 6 x minus 4 end exponent Apply the property open parentheses b to the power of x close parentheses to the power of y equals b to the power of x y end exponent.
x minus 3 equals 6 x minus 4 Use the one-to-one property.
short dash 5 x minus 3 equals short dash 4
short dash 5 x equals short dash 1
x equals 1 fifth 		Solve for x.

The solution is x equals 1 fifth. When substituting into the original equation, it checks.

try it
Consider the equation 5 to the power of x plus 8 end exponent equals 25 to the power of x minus 3 end exponent.
Solve the equation for x.
The bases are not the same, but since 25 equals 5 squared comma we can write the right-hand side as a power of 5.

5 to the power of x plus 8 end exponent equals open parentheses 5 squared close parentheses to the power of x minus 3 end exponent Replace 25 with 5 squared.
5 to the power of x plus 8 end exponent equals 5 to the power of 2 x minus 6 end exponent Use properties of exponents to simplify the right-hand side. Note that the exponents are multiplied, and 2 open parentheses x minus 3 close parentheses equals 2 x minus 6.
x plus 8 equals 2 x minus 6 Since the bases are equal, set the exponents equal.
8 equals x minus 6 Subtract x from both sides.
14 equals x Add 6 to both sides. This is the solution.

The solution to the equation is x equals 14.

There are even times when it is difficult to write one base as the power of the other, but both bases can be converted to the same base.

EXAMPLE

Solve the equation 9 to the power of x plus 2 end exponent equals 27 to the power of x minus 1 end exponent.

It is more difficult to write 27 as a power of 9. Notice that both 9 and 27 are integer powers of 3.

9 to the power of x plus 2 end exponent equals 27 to the power of x minus 1 end exponent This is the original equation.
open parentheses 3 squared close parentheses to the power of x plus 2 end exponent equals open parentheses 3 cubed close parentheses to the power of x minus 1 end exponent Rewrite 9 equals 3 squared and 27 equals 3 cubed.
3 to the power of 2 open parentheses x plus 2 close parentheses end exponent equals 3 to the power of 3 open parentheses x minus 1 close parentheses end exponent Apply the property open parentheses b to the power of x close parentheses to the power of y equals b to the power of x y end exponent.
3 to the power of 2 x plus 4 end exponent equals 3 to the power of 3 x minus 3 end exponent Simplify both exponents.
2 x plus 4 equals 3 x minus 3 Apply the one-to-one property.
4 equals x minus 3
7 equals x 		Solve for x.

The solution is x equals 7. When substituting into the original equation, it checks.

watch
To see the solution of 8 to the power of x minus 1 end exponent equals open parentheses 1 fourth close parentheses to the power of x plus 2 end exponent comma check out the video below.

try it
Consider the equation 64 to the power of x plus 1 end exponent equals 16 to the power of 3 x minus 1 end exponent.
Solve the equation for x.
Note that the bases aren’t equal, and it is not straightforward to write 64 as a power of 16. But, both bases are powers of 4.

open parentheses 4 cubed close parentheses to the power of x plus 1 end exponent equals open parentheses 4 squared close parentheses to the power of 3 x minus 1 end exponent Write 64 equals 4 cubed and 16 equals 4 squared.
4 to the power of 3 x plus 3 end exponent equals 4 to the power of 6 x minus 2 end exponent Use exponent properties to simplify the exponents. Since the exponents are multiplied, note that 3 open parentheses x plus 1 close parentheses equals 3 x plus 3 and 2 open parentheses 3 x minus 1 close parentheses equals 6 x minus 2.
3 x plus 3 equals 6 x minus 2 Since the bases are equal, the exponents are also equal.
3 equals 3 x minus 2 Subtract 3 x from both sides.
5 equals 3 x Add 2 to both sides.
5 over 3 equals x Divide both sides by 3. This is the solution.

The solution to the equation is x equals 5 over 3.

summary
In this lesson, you learned that solving exponential equations with like bases requires the use of the one-to-one property of exponential functions. This property can also be used when solving exponential equations by changing one or both bases, so that they have the same base.

SOURCE: THIS WORK IS ADAPTED FROM PRECALCULUS BY JAY ABRAMSON. ACCESS FOR FREE AT OPENSTAX.ORG/BOOKS/PRECALCULUS/PAGES/1-INTRODUCTION-TO-FUNCTIONS

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