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# Inverse Properties of Logarithms

## Simplify expressions using two properties of inverse logs

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Inverse Properties of Logarithmic Functions

If you continue to study mathematics into college, you may take a course called Differential Equations. There you will learn that the solution to the differential equation is the general function . What is the inverse of this function?

### Inverse Properties of Logarithms

By the definition of a logarithm, it is the inverse of an exponent. Therefore, a logarithmic function is the inverse of an exponential function. Recall what it means to be an inverse of a function. When two inverses are composed, they equal . Therefore, if and , then:

and

These are called the Inverse Properties of Logarithms.

Let's solve the following problems. We will use the Inverse Properties of Logarithms.

1. Find .

Using the first property, we see that the bases cancel each other out.

Here, and the natural log cancel out and we are left with .

1. Find .

We will use the second property here. Also, rewrite 16 as .

1. Find the inverse of .

Change to . Then, switch and .

Now, we need to isolate the exponent and take the logarithm of both sides. First divide by 2.

Recall the Inverse Properties of Logarithms from earlier in this concept. ; applying this to the right side of our equation, we have . Solve for .

Therefore, is the inverse of .

### Examples

#### Example 1

Earlier, you were asked to find the inverse of  .

Switch x and y in the function and then solve for y.

Therefore, the inverse of is .

#### Example 2

Simplify .

Using the first inverse property, the log and the base cancel out, leaving as the answer.

#### Example 3

Simplify .

Using the second inverse property and changing 81 into we have:

#### Example 4

Find the inverse of .

### Review

Use the Inverse Properties of Logarithms to simplify the following expressions.

Find the inverse of each of the following exponential functions.

To see the Review answers, open this PDF file and look for section 8.6.

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### Vocabulary Language: English

Inverse Properties of Logarithms

The inverse properties of logarithms are $\log_b b^x=x$ and $b^{\log_b x}=x, b \ne 1$.