**Learning Objectives**

After completing this tutorial, you should be able to:

- Solve logarithmic equations.

**Introduction**

In this tutorial I will step you through how to solve equations that
have logarithmic expressions. In these equations, you will notice
that the variable that we are solving for is inside the log expressions.
We will rewrite the log equation exponentially using the definition of
logs to help us get the *x* outside of the log.
If you need a review on the definition of log functions, feel free to go
to **Tutorial 43: Logarithmic Functions**.
Ready, set, GO!!!!!

** Tutorial**

**Solving a Logarithmic Equation of the Form **

Get your log on one side everything outside of the log on the other
side of your equation using inverse operations.

Also use properties of logs to write it so that there is only one log.
If you need a review on properties of logs feel free to go to **Tutorial
44: Logarithmic Properties**.

A reminder that the definition of logarithms is the logarithmic function
with base *b*, where *b* > 0 and *b*0,
and is defined as if and only if .

If you need a review on the definition of log functions, feel free to
go to **Tutorial 43: Logarithmic
Functions**.

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

This is already done for us in this problem.

Remember that when there is no base written on a log, that means it
is the common log, or log base 10. If you need a review on common
logs feel free to go to **Tutorial
43: Logarithmic Functions**.

***Set 1st factor = 0 **

***Set 2nd factor = 0 **

**Final answer: x = 4.**

***Multiply both sides by LCD of x + 2**

The equation that we had to solve in step 3 had a rational expression
in it. If you need a review on solving equations with rational expressions
feel free to go to **Tutorial 15: Equations
with Rational Expressions.**

***one log is isolated**

Remember that when you have ln, that means it is the natural log, or
log base *e*. If you need a review on natural
logs feel free to go to **Tutorial
43: Logarithmic Functions**.

***Solve for x**

***Use the calculator to find e squared**

The equation that we had to solve in step 3 had a square root in it.
If you need a review on solving equations with radicals feel free to go
to **Tutorial 19: Radical Equations
and Equations Involving Rational Exponents.**

** 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 - 1d:Solve the logarithmic 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/solvelog.htm**

This webpage will help you with solving logarithmic equations.

**http://www.sosmath.com/algebra/solve/solve8/s81/s81.html**

This webpage gives an example of solving a logarithmic 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.