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Solving Exponential Equations That Contain Like Bases

Author: Sophia

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:

Solving Exponential Equations With Like Bases
Solving Exponential Equations by Changing One or Both Bases

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.

		This is the original equation.
		Apply the one-to-one property.
		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.

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.

		This is the original equation.
		Rewrite so that the bases on both sides are equal.
		Apply the property
		Use the one-to-one property.
		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.

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.

		This is the original equation.
		Rewrite and
		Apply the property
		Simplify both exponents.
		Apply the one-to-one property.
		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

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.

Video Transcription

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Hi. I'm glad you joined me today in solving an exponential equation where the bases can be written in terms of the same base. So here we have to solve 8 to the x minus 1 power is equal to 1/4 to the x plus 2 power. Now when we're trying to solve exponential equations, the first thing we look for is if we have the same base on both sides. And if not, do we have numbers that can be written as the same base?

Well, 8 can be written as 2 cubed. And 1/4 can be written as 2 to the negative 2 power. So we can write the left side as 2 cubed to the x minus 1. And on the right side, we can write it as 2 to the negative 2 power to the x plus 2.

Now recall with exponential expressions, if I have an exponential expression to get another power, to simplify that, we keep the base and we multiply the exponents. So on the left, we have 2 to the 3 times x minus 1 power, keeping the base and multiplying the exponents. And on the right, we have 2 to the negative 2 times x plus 2 power.

Go ahead and distribute in the exponents. We have 2 to the 3x minus 3 is equal to 2 to the negative 2x minus 4. Now exponentials are one to one. So that means if we have equality and the bases are equal, the exponents have to be equal. So this 3x minus 3 has to equal negative 2x minus 4.

Solving for x will add 2x to both sides. So 5x minus 3 is equal to negative 4. Add 3 to both sides. 5x is equal to negative 1. And then divide both sides by 5. We get x is equal to negative 1/5.

And I'll leave it to you to check that in the original equation. But x equals negative 1/5 is the solution to our equation 8 to the x minus 1 is equal to 1/4 to the x plus 2.

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.

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