**Learning Objectives**

After completing this tutorial, you should be able to:

- Solve exponential equations.

**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.

** Tutorial**

**Solving Exponential Equations, **

**where x is in the exponent, BUT**

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

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.

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.

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

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.

This is already done for us in this problem.

***Use the calculator to find ln 50**

***ln e is 1**

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

***Exponential expression isolated**

***Inverse of add 1 is sub. 1**

***Use the calculator to find ln 2.4 and ln 10**

***Exponential expression isolated**

***Use the calculator to find ln 21 and ln 2**

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 .

***Set the 1st factor = 0**

***Isolate the exponential expression**

***Set the 2nd factor = 0**

***Isolate the exponential expression**

**So there is only one equation that we can solve **.

***Use the calculator to find ln 4**

***ln e = 1**

** 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.

Practice Problems 1a - 1c:Solve the given exponential equation. Round your answer to two decimal places.

** Need Extra Help on these Topics?**

**The following are webpages that can assist you in the topics that were covered on this page**.

**http://www.purplemath.com/modules/solvexpo.htm**

This webpage helps you with exponential equations.

**http://www.sosmath.com/algebra/solve/solve7/s72/s72.html**

This webpage gives an example of solving an exponential equation.

**http://www.sosmath.com/algebra/solve/solve7/s73/s73.html**

This webpage gives an example of solving an exponential equation.

**Go to Get
Help Outside the
Classroom found in Tutorial 1: How to Succeed in a Math Class for
some
more suggestions.**

Last revised on March 24, 2011 by Kim Seward.

All contents copyright (C) 2002 - 2011, WTAMU and Kim Seward. All rights reserved.