In this explainer, we will learn how to solve exponential equations using the properties of exponents.

Let’s start by looking at some examples of exponential equations. One is , another is , and a third is . Notice that the variable occurs in the exponent on the left side of the first equation, the variable occurs in the exponent on both sides of the second equation, and the variable occurs in the exponent on the right side of the third equation. These three examples of exponential equations lead us to the following definition.

### Definition: Exponential Equation

An exponential equation is an equation in which a variable is used in one exponent or more.

To determine the solution set of an exponential equation, it is often useful to rewrite it so that each side
consists of the same **base** raised to a **power**. In order to do this, we often need to recall some of the
rules of exponents below.

### Properties: Rules of Exponents

Notice that the rules in the left column each involve operations with two exponents.

- The
**product rule**states that when multiplying exponential expressions with the same base, we keep the base and find the sum of the exponents. We would use the**product rule**to find that . - Similarly, the
**quotient rule**states that when dividing exponential expressions with the same base, we keep the base and find the difference of the exponents. We would use the**quotient rule**to find that . - The
**power rule**states that when raising a power of a base to another power, we keep the base and find the product of the exponents. We would use the**power rule**to find that .

The rules in the right column allow us to simplify or rewrite an expression involving a base raised to a certain type of power.

- The
**zero exponent rule**states that any base raised to the power 0 is equal to 1. We would use the**zero exponent rule**to find that . - The
**negative exponent rule**states that any base raised to a negative power is equal to 1 over the base raised to the exponent’s additive inverse. We would use the**negative exponent rule**to find that . - Finally, the
**fractional exponent rule**states that any base raised to a fractional power with a numerator of 1 is equal to a root of the base. The root’s degree is the exponent’s denominator.

We would use the**fractional exponent rule**to find that .

Now, let’s look at some examples of how these rules are used when solving exponential equations.

### Example 1: Finding the Solution Set of Exponential Equations

Given that , find the value of .

### Answer

In order to find the value of , we should begin by rewriting the equation so that both sides consist of the same base raised to a power. The left side of the equation already is a base raised to a power—the base 2 is being raised to the power . Thus, we must determine if the right side of the equation can also be written as the base 2 raised to a power. If it can, then we will not have to rewrite the equation’s left side. We will only need to rewrite the right side.

Consider the first five positive integer powers of 2:

We can see that 32 can be rewritten as the base 2 raised to the power 5. This means that we can leave the left side of the equation unchanged and rewrite the equation as

By inspection, we can now find the value of , since the exponents on both sides of the equation must be the same. The value of is 5.

### Check

We can check our answer by substituting 5 into the original equation for . Doing so gives , which can be rewritten as , so our answer must be correct.

Note that we can also find the value of by taking the fifth root of both sides of the original equation, although this method is not nearly as straightforward and requires using two of the rules of exponents. First, we get , which can be rewritten as

The **fractional exponent rule** allows us to rewrite as
, so the equation becomes
and the **power rule** allows us to rewrite as
, so the equation then becomes

Rewriting 2 as then gives us . Equating the exponents gives us , and multiplying both sides of by 5 gives us , the same answer that we got before.

In the next example, we will evaluate an exponential expression after solving a pair of exponential equations.

### Example 2: Evaluating Exponential Expressions after Solving Exponential Equations

Given that , find the value of .

### Answer

To find the value of , we should start by rewriting as two separate equations:

Let’s consider the equation first. We can begin to solve it by rewriting it so that both sides consist of the same base raised to a power. The left side already is a base raised to a power – the base 8 is being raised to the power . Thus, we must determine if the right side can also be written as the base 8 raised to a power. If it can, then we will only need to rewrite this side, and not the left side.

Since , we can leave the left side of the equation unchanged and rewrite the equation as

By then equating the exponents, we can conclude that .

Now let’s consider the equation . Again, we should rewrite it so that both sides consist of
the same base raised to a power. Since and ,
we can rewrite the equation as
with each side having the base 2. The **power rule** of Exponents allows us to rewrite
as and as , so the equation becomes

Since the exponents on both sides of the equation must be the same, we know that .

Recall that we got as the solution to the equation . Because we already know the value of , we can substitute it into the equation to get , or . Dividing both sides of by 2 then gives us .

Note that we can also find the value of by rewriting as the equations

To solve , we can leave the left side of the equation unchanged and use the fact that to rewrite the equation as

By equating the exponents, we can conclude that . This is the same value of that we arrived at previously.

Since we now know the values of both and , we can substitute them into the expression and simplify: . The value of is 5.

Now, let’s consider a problem involving **binomial exponents**.

### Example 3: Finding the Solution Set of an Exponential Equation Involving Binomial Exponents

Find the value of for which .

### Answer

We have been asked to find the value of for which . On the left side of this equation, the base 81 is being raised to the power . Since has two terms, it is the binomial exponent.

On the right side, the base is being raised to the power . Since both 81 and are powers of 3, the easiest approach would be to rewrite the equation so that 3 is the base on both sides.

Doing so, we get

We can then use the **power rule** of exponents to rewrite the equation as
and distribute in the exponent on the left side to get

Setting the exponents equal to each other then gives us the equation and allows us to solve for . We would subtract 20 from both sides to get and then add to both sides to get

Finally, by dividing both sides by 5, we can find the value of for which . The value of is .

The problem that follows is similar to the one we have just looked at, but the equation we will solve includes two binomial exponents instead of one.

### Example 4: Finding the Solution Set of an Exponential Equation Involving Binomial Exponents

Find the value of for which . Give your answer to the nearest tenth.

### Answer

We can find the value of for which by first rewriting the equation so that each side consists of the same base raised to a power. In this case, the power on each side will be represented by a binomial exponent.

Since , we can rewrite the equation as

Now both sides of the equation have the base 2, so there is no need to rewrite the right side.

Next, we can use the **power rule** of exponents to rewrite
as , giving us the equation
and we can then distribute the 3 in the exponent on the left side to get

Equating the two binomial exponents gives us the equation allowing us to solve for . First, subtracting from both sides gives us the equation

Then, subtracting 6 from both sides, we get

Finally, by dividing both sides by 2, we can find the value of for which . The value of is .

### Check

Now let’s check our answer. Substituting into the equation for gives us , and after simplifying the binomial exponent on each side, we get . Since and , we can rewrite the equation as . Thus, our value of must be correct.

Sometimes absolute value symbols are used in exponential equations. In the next example, we will look at such an equation.

### Example 5: Solving Exponential Equations Involving Absolute Value Using Laws of Exponents

Find the solution set of .

### Answer

Let’s begin by rewriting the equation so that the same base is raised to a power on each side. We can use the fact that to rewrite the equation as

Now that there is a base of 2 on each side, we can use the **power rule** of exponents to replace
with ,
giving us the equation
and after distributing the 3 in the exponent on the right side, we get

Since the exponents must be equal to each other, we know that the equation must hold true.

The equation is an absolute value equation, so we must consider two different cases: when is positive and when it is negative. When it is positive, we get the equation , and when it is negative, we get the equation . Each of these equations can be solved for as shown below:

Solving the equations shows that could be equal to 0 or . However, we must substitute each value into the equation and simplify as follows to be sure:

Since is false and is true, we know that only , and not 0, is in the solution set of the equation . Therefore, the solution set is .

Finally, let’s look at one more example of an exponential equation involving binomial exponents.

### Example 6: Finding the Solution Set of an Exponential Equation Involving Binomial Exponents

Determine the solution set of .

### Answer

Notice that the exponent on both sides of the equation is . Since the exponent on one side is the same as that on the other, we know that one element of the equation’s solution set must be 6. This is because a value of 6 for would result in the same base being raised to the same power on each side. Substituting 6 into the equation for gives us and partially simplifying gives

Suppose, however, that we substituted for instead. After partially simplifying, we would get

Since is an even number, both and 6 raised to this power would give the same result. This means that another element of the solution set must be .

We must also remember that the **zero exponent rule** states that any base raised to the power 0 is
equal to 1. Thus, any values of that result in the exponent
being equal to 0 would also be in the solution set to the equation .
The value of the base on the left side of the equation would not matter in this case. We can solve the equation
as follows to find these additional solution set elements:

Thus, two additional elements of the solution set are 8 and , so we now know the complete solution set of . The solution set is .

### Check

We can check that 8 and are indeed values of that make the equation true by substituting them into the equation and simplifying. Substituting 8 for gives which can be simplified to

Substituting for gives
which can be simplified to
Using the **zero exponent rule**, both and
can be simplified to , which is true. Thus, both 8 and are, in fact,
elements of the solution set.

Now, let’s finish by recapping some key points.

### Key Points

- An exponential equation is an equation in which a variable is used in one exponent or more.
- To determine the solution set of an exponential equation, it is often useful to rewrite it so that each side consists of the same base raised to a power.
- To rewrite an exponential equation so that each side consists of the same base raised to a power, it is sometimes necessary to use one or more of the rules of exponents.
- An exponential rule that is commonly used when solving exponential equations is the
**power rule**, or . It states that when raising a power of a base to another power, we keep the base and find the product of the exponents. - Other rules of exponents include the
**product rule**, the**quotient rule**, the**zero exponent rule**, the**negative exponent rule**, and the**fractional exponent rule**. - A solution to an exponential equation should be substituted back into the original equation to check that it is correct.