# 8.11: Solving Logarithmic Equations

**Practice**Solving Logarithmic Equations

"I'm thinking of another number," you tell your best friend. "The number I'm thinking of satisfies the equation ." What number are you thinking of?

### Guidance

A logarithmic equation has the variable within the log. To solve a logarithmic equation, you will need to use the inverse property, , to cancel out the log.

#### Example A

Solve .

**Solution:** There are two different ways to solve this equation. The first is to use the definition of a logarithm.

The second way to solve this equation is to put everything into the exponent of a 2, and then use the inverse property.

Make sure to check your answers for logarithmic equations. There can be times when you get an extraneous solution.

#### Example B

Solve .

**Solution:** First, add 5 to both sides and then divide by 3 to isolate the natural log.

Recall that the inverse of the natural log is the natural number. Therefore, everything needs to be put into the exponent of in order to get rid of the log.

Checking the answer, we have

#### Example C

Solve

**Solution:** Condense the left-hand side using the Product Property.

Now, put everything in the exponent of 10 and solve for .

Now, check both answers.

-4 is an extraneous solution. In the step , we cannot take the log of a negative number, therefore -4 is not a solution. 5 is the only solution.

**Intro Problem Revisit** We can rewrite as and solve for *x*.

Therefore, the number you are thinking of is 100.

### Guided Practice

Solve the following logarithmic equations.

1.

2.

3.

#### Answers

1. Isolate the log and put everything in the exponent of 3.

2. Condense the left-hand side using the Quotient Rule and put everything in the exponent of .

Checking our answer, we get , which does not work because the first natural log is of a negative number. Therefore, there is no solution for this equation.

3. Multiply both sides by 2 and put everything in the exponent of a 5.

### Practice

Use properties of logarithms and a calculator to solve the following equations for . Round answers to three decimal places and check for extraneous solutions.

### Image Attributions

## Description

## Learning Objectives

Here you'll learn how to solve a logarithmic equation with any base.