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Solving an Exponential Equation Using the Base

Author: Sophia
PDF Version

what's covered
In this lesson, you will learn how to solve an exponential equation by rewriting the base. Specifically, this lesson will cover:

Table of Contents

1. Solving an Exponential Equation with the Same Base

There are several ways to solve an exponential equation. The way we will approach this problem first is by analyzing the bases involved in the equations. In a later section, we will utilize the inverse of an exponent, the logarithm, to solve these types of equations.

EXAMPLE

If 6 to the power of 2 x plus 9 end exponent equals 6 to the power of short dash 5 x plus 2 end exponent, then what is the value of x?

In both exponential expressions, the base is 6. We can say 6 to the power of x equals 6 to the power of y is true, only if x equals y. So, in our example, this means that the quantities in the exponent of 6 are the same on both sides of the equation. Therefore 2 x plus 9 and short dash 5 x plus 2 must be equal quantities. We can create an equivalent equation that is actually linear in nature, and solve for x:

6 to the power of 2 x plus 9 end exponent equals 6 to the power of short dash 5 x plus 2 end exponent The equation.
2 x plus 9 equals short dash 5 x plus 2 Since both sides of the exponential equation has the same bases of 6, set exponents equal to each other and solve like a linear equation.
7 x plus 9 equals 2 Add 5 x to both sides of the equation.
7 x equals short dash 7 Subtract 9 from both sides.
x equals short dash 1 Divide both sides by 7.

big idea
If the bases are the same in exponential equations, we can set the exponents equal to each other, and isolate the variable as we normally do with other equations.

try it
Consider the equation 2 to the power of x plus 6 end exponent equals 2 to the power of short dash 2 x minus 3 end exponent.
What is the value of x?
2 to the power of x plus 6 end exponent equals 2 to the power of short dash 2 x minus 3 end exponent The equation.
x plus 6 equals short dash 2 x minus 3 Since both sides of the exponential equation has the same bases of 2, set exponents equal to each other and solve like a linear equation.
3 x plus 6 equals short dash 3 Add 2 x to both sides of the equation.
3 x equals short dash 9 Subtract 6 from both sides.
x equals short dash 3 Divide both sides by 3.

2. Rewriting the Base

When we are working with exponential equations in which the base numbers are not the same, it may appear as though we cannot solve using the strategy described in the section above. However, by closely examining the base numbers, we may be able to rewrite one or more of the bases in order to create an equivalent equation with common bases. If we can do this, we can solve the equation using a similar strategy as before.

hint
When we use this strategy, we will need to apply the power of a power property of exponents. This property allows us to multiply exponents in cases where a base number is raised to an exponent power and then raised to an exponent power again.

formula to know
Power of a Power Property of Exponents
open parentheses a to the power of n close parentheses to the power of m equals a to the power of n m end exponent

EXAMPLE

If 4 to the power of x plus 3 end exponent equals 8 to the power of x minus 1 end exponent, then what is the value of x?

At first glance, it may appear as though we cannot solve this equation using our strategy from before. However, we notice that both 4 and 8 are powers of 2. That is, 4 is the same as 22 and 8 is the same as 23. Let's make these substitutions in our equation by rewriting each base.

4 to the power of x plus 3 end exponent equals 8 to the power of x minus 1 end exponent The equation.
open parentheses 2 squared close parentheses to the power of x plus 3 end exponent equals open parentheses 2 cubed close parentheses to the power of x minus 1 end exponent Exponential equation with different bases, so rewrite the base: 4 can be rewritten as 2 squared and 8 can be rewritten as 2 cubed.
2 to the power of 2 x plus 6 end exponent equals 2 to the power of 3 x minus 3 end exponent Now we have the same bases and can solve as before. We can multiply the two exponents on each side of the equation using the power of powers property of exponents.
2 x plus 6 equals 3 x minus 3 Same base so we can set exponents equal to each other
2 x plus 9 equals 3 x Add 3 to both sides.
9 equals x Subtract 2 x from both sides.

When x equals 9, the two sides of the equation are equal. We can test this by plugging 9 in for x.

4 to the power of x plus 3 end exponent equals 8 to the power of x minus 1 end exponent The equation.
4 to the power of 9 plus 3 end exponent equals 8 to the power of 9 minus 1 end exponent Plug in 9 for x.
4 to the power of 12 equals 8 to the power of 8 Evaluate operations in exponents.
16 comma 777 comma 216 equals 16 comma 777 comma 216 Evaluate exponents. This is a true statement, ensuring that 9 is the solution to x.

summary
In this lesson, you learned about one method for solving exponential equations. When solving an exponential equation with the same base, set the exponents equal to each other and create an equivalent equation that is linear in nature, and solve for x. If the bases are not the same, you learned that you will need to rewrite the base in order to create an equivalent equation with common bases. Then, you can set the exponents equal to each other and solve like a linear equation.

Best of luck in your learning!

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