How To Get A Variable Out Of An Exponent

College Algebra
Tutorial 45: Exponential Equations

$desk$ Learning Objectives

After completing this tutorial, you should be able to:

Solve exponential equations.

$desk$ Introduction

In this tutorial I will walk you through how to solve equations that have exponential expressions. In these equations, you will notice that the variable that we are solving for is in the exponent. We are use to seeing the variable in the base. We will using inverse operations like we do in linear equations, the inverse operation we will be using here is logarithms. If you need a review on the definition of log functions, feel free to go to Tutorial 43: Logarithmic Functions. If you need a review on log properties, feel free to go to Tutorial 44: Logarithmic Properties. I think you are ready to get started.

$desk$ Tutorial

Solving Exponential Equations,
where x is in the exponent, BUT
the bases DO NOT MATCH.

Step 1: Isolate the exponential expression.

Get your exponential expression on one side everything outside of the exponential expression on the other side of your equation.

Step 2: Take the natural log of both sides.

The inverse operation of an exponential expression is a log. Make sure that you do the same thing to both sides of your equation to keep them equal to each other.

Step 3: Use the properties of logs to pull the x out of the exponent.

Now that the variable is out of the exponent, solve for the variable using inverse operations to complete the problem.

Special Note:
The only way we can get that variable out of the exponent, when the bases don't match up, is to use logs. The 3rd step allows us to do this. When solving an equation, it doesn't matter what you do to the equation as long as you do the same thing to both sides - this keeps both sides equal. Also, the reason we take the natural log of both sides is because we have the natural log key on the calculator - so we would be able to find a value of it in the end.

$notebook$ Example 1 : Solve the exponential equation $example 1a$ . Round your answer to two decimal places.

This is already done for us in this problem.

$example 1b$

*Take the natural log of BOTH sides

$example 1d$

*Inverse of mult. by 3ln e is to divide by 3ln e

*Use the calculator to find ln 50
*ln e is 1

$notebook$ Example 2 : Solve the exponential equation $example 2a$ . Round your answer to two decimal places.

$example 2b$

*Inverse of mult. by 5 is to divide by 5

*Exponential expression isolated

$example 2c$

*Take the natural log of BOTH sides

$example 2e$

*Inverse of mult. by ln 10 is to divide by ln 10

*Inverse of add 1 is sub. 1

*Use the calculator to find ln 2.4 and ln 10

$notebook$ Example 3: Solve the exponential equation $example 3a$ . Round your answer to two decimal places.

$example 3b$

*Inverse of add 4 is sub. 4

*Exponential expression isolated

$example 3c$

*Take the natural log of BOTH sides

$example 3e$

*Inverse of mult. by .2ln 2 is to divide by .2ln 2

*Use the calculator to find ln 21 and ln 2

$notebook$ Example 4: Solve the exponential equation $example 4h$ . Round your answer to two decimal places.

Notice how we have two exponential terms that have different exponents. We wouldn't be able to isolate both. We will have to figure out another way to rewrite it so we can continue with the steps.

Note how we have a trinomial and that e to the 2 x is e to the x squared. This means it is quadratic in from. So we can factor it just like a trinomial of the form $example 4b$ .

$example 4i$

*Set the 1st factor = 0
*Isolate the exponential expression

*Set the 2nd factor = 0
*Isolate the exponential expression

Note that since e is a positive base, no matter what the exponent is on x , this exponential expression CANNOT equal -2.

So there is only one equation that we can solve $example 4d$ .

$example 4e$

*Take the natural log of BOTH sides

$example 4g$

*Inverse of mult. by ln e is to divide by ln e

*Use the calculator to find ln 4
*ln e = 1

$desk$ Practice Problems

These are practice problems to help bring you to the next level. It will allow you to check and see if you have an understanding of these types of problems. Math works just like anything else, if you want to get good at it, then you need to practice it. Even the best athletes and musicians had help along the way and lots of practice, practice, practice, to get good at their sport or instrument. In fact there is no such thing as too much practice.

To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that problem. At the link you will find the answer as well as any steps that went into finding that answer.