"I'm thinking of a number," you tell your best friend. "The number I'm thinking of satisfies the equation

### Solving Logarithm Equations

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

Let's solve the following logarithmic equations.

log2(x+5)=9

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.

3ln(−x)−5=10

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

Checking the answer, we have

log5x+log(x−1)=2

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

### Examples

#### Example 1

Earlier, you were asked to determine what number you are thinking of if the number satisfies the equation

We can rewrite *x*.

Therefore, the number you are thinking of is 100.

#### Example 2

Solve:

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

#### Example 3

Solve:

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

Checking our answer, we get

#### Example 4

Solve:

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

### Review

Use properties of logarithms and a calculator to solve the following equations for

log2x=15 log12x=2.5 log9(x−5)=2 log7(2x+3)=3 8ln(3−x)=5 4log33x−log3x=5 log(x+5)+logx=log14 2lnx−lnx=0 3log3(x−5)=3 23log3x=2 5logx2−3log1x=log8 2lnxe+2−lnx=10 2log6x+1=log6(5x+4) 2log12x+2=log12(x+10) 3log23x−log2327=log238

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 8.11.