## Using Like Bases to Solve Exponential Equations

Getting the variable out of the exponent can be tricky, but you will learn the basics in this section.

### Using Like Bases to Solve Exponential Equations

##### Learning Objectives

In this section, you will:

• Use like bases to solve exponential equations.
• Use logarithms to solve exponential equations.
• Use the definition of a logarithm to solve logarithmic equations.
• Use the one-to-one property of logarithms to solve logarithmic equations.
• Solve applied problems involving exponential and logarithmic equations.

Figure 1 Wild rabbits in Australia. The rabbit population grew so quickly in Australia that the event became known as the "rabbit plague".

In 1859, an Australian landowner named Thomas Austin released 24 rabbits into the wild for hunting. Because Australia had few predators and ample food, the rabbit population exploded. In fewer than ten years, the rabbit population numbered in the millions.

Uncontrolled population growth, as in the wild rabbits in Australia, can be modeled with exponential functions. Equations resulting from those exponential functions can be solved to analyze and make predictions about exponential growth. In this section, we will learn techniques for solving exponential functions.

#### Using Like Bases to Solve Exponential Equations

The first technique involves two functions with like bases. Recall that the one-to-one property of exponential functions tells us that, for any real numbers $b$, $S$, and $T$, where $b > 0, b ≠1, b^S=b^T$ if and only if $S=T$.

In other words, when an exponential equation has the same base on each side, the exponents must be equal. This also applies when the exponents are algebraic expressions. Therefore, we can solve many exponential equations by using the rules of exponents to rewrite each side as a power with the same base. Then, we use the fact that exponential functions are one-to-one to set the exponents equal to one another, and solve for the unknown.

For example, consider the equation $3^{4x−7}=\dfrac{3^{2x}}{3}$. To solve for $x$, we use the division property of exponents to rewrite the right side so that both sides have the common base, $3$. Then we apply the one-to-one property of exponents by setting the exponents equal to one another and solving for $x$:

$3^{4x-7} = \dfrac{3^{2x}}{3}$
$3^{4x-7} = \dfrac{3^{2x}}{3^1}$

Rewrite $3$ as $3^1$.

$3^{4x-7} = 3^{2x-1}$

Use the division property of exponents.

$4x-7= 2x-1$

Apply the one-to-one property of exponents.

$2x= 6$

Subtract $2x$ and add $7$ to both sides.

$x= 3$

Divide by $3$.

##### Using the One-to-One Property of Exponential Functions to Solve Exponential Equations

For any algebraic expressions $Sand T$, and any positive real number $b≠1$,

$b^S=b^T$ if and only if $S=T$

##### How To

Given an exponential equation with the form $b^S=b^T$, where $S$ and $T$ are algebraic expressions with an unknown, solve for the unknown.

1. Use the rules of exponents to simplify, if necessary, so that the resulting equation has the form $b^S=b^T$.
2. Use the one-to-one property to set the exponents equal.
3. Solve the resulting equation, $S=T$, for the unknown.

##### Example 1

Solving an Exponential Equation with a Common Base
Solve $2^{x−1}=2^{2x−4}$.

Solution

$2^{x−1}=2^{2x−4}$

The common base is 2.

$x−1=2x−4$

By the one-to-one property the exponents must be equal.

$x= 3$

Solve for x.

##### Try It #1

Solve $5^{2x}=5^{3x+2}$.

#### Rewriting Equations So All Powers Have the Same Base

Sometimes the common base for an exponential equation is not explicitly shown. In these cases, we simply rewrite the terms in the equation as powers with a common base, and solve using the one-to-one property.

For example, consider the equation $256=4^{x−5}$. We can rewrite both sides of this equation as a power of $2$. Then we apply the rules of exponents, along with the one-to-one property, to solve for $x$:

$256=4^{x−5}$
$2^8=(2^2)^{x−5}$

Rewrite each side as a power with base 2.

$2^8=2^{2x−10}$ Use the one-to-one property of exponents.
$8 = 2x-10$
Apply the one-to-one property of exponents.
$18 = 2x$
$x=9$ Divide by 2.

##### How To

Given an exponential equation with unlike bases, use the one-to-one property to solve it.

1. Rewrite each side in the equation as a power with a common base.
2. Use the rules of exponents to simplify, if necessary, so that the resulting equation has the form $b^S=b^T$.
3. Use the one-to-one property to set the exponents equal.
4. Solve the resulting equation, $S=T$, for the unknown.

##### Example 2

Solving Equations by Rewriting Them to Have a Common Base
Solve $8^{x+2}=16^{x+1}$.

Solution

Sometimes the common base for an exponential equation is not explicitly shown. In these cases, we simply rewrite the terms in the equation as powers with a common base, and solve using the one-to-one property.

For example, consider the equation $256=4^{x−5}$. We can rewrite both sides of this equation as a power of $2$. Then we apply the rules of exponents, along with the one-to-one property, to solve for $x$:

$8^{x+2}=16^{x+1}$
$(2^3)^{x+2} = (2^4)^{x+1}$

Write $8$ and $16$ as powers of $2$.

$2^{3x+6} = 2^{4x+4}$
To take a power of a power, multiply exponents.
$3x+6 = 4x+4$
Use the one-to-one property to set the exponents equal.
$x=2$ Solve for $x$.

##### Try It #2

Solve $5^{2x}=25^{3x+2}$.

##### Example 3

Solving Equations by Rewriting Roots with Fractional Exponents to Have a Common Base

Solve $2^{5x}=\sqrt{2}$.

Solution

$2^{5x}=2\dfrac{1}{2}$

Write the square root of $2$ as a power of $2$.

$x=\dfrac{1}{2}$

Use the one-to-one property.

$x=\dfrac{1}{10}$

Solve for $x$.

Solve 5x=5–√.

##### Q&A

Do all exponential equations have a solution? If not, how can we tell if there is a solution during the problem-solving process?

No. Recall that the range of an exponential function is always positive. While solving the equation, we may obtain an expression that is undefined.

##### Example 4

Solving an Equation with Positive and Negative Powers
Solve $3^{x+1}=−2$.

Solution

This equation has no solution. There is no real value of $x$ that will make the equation a true statement because any power of a positive number is positive.

Analysis
Figure 2 shows that the two graphs do not cross so the left side is never equal to the right side. Thus the equation has no solution.
Graph of 3^(x+1)=-2 and y=-2. The graph notes that they do not cross.

Figure 2

##### Try It #4

Solve $2^x=−100$.

Source: Rice University, https://openstax.org/books/college-algebra/pages/6-6-exponential-and-logarithmic-equations