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.

- 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 .

- Find .

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

- 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.

### Answers for Review Problems

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