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?

### Guidance

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 (see the
*
Inverse of a Function
*
concept), they equal
. Therefore, if
and
, then:

and

These are called the Inverse Properties of Logarithms.

#### Example A

Find:

a)

b)

**
Solution:
**
For each of these examples, we will use the Inverse Properties.

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

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

#### Example B

Find

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

#### Example C

Find the inverse of .

**
Solution:
**
See the
*
Finding the Inverse
*
concept for the steps on how to find the inverse.

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

Therefore, is the inverse of .

**
Intro Problem Revisit
**
Switch
*
x
*
and
*
y
*
in the function
and then solve for
*
y
*
.

Therefore, the inverse of is .

### Guided Practice

1. Simplify .

2. Simplify .

3. Find the inverse of .

#### Answers

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

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

3. Follow the steps from Example C to find the inverse.

### Vocabulary

- Inverse Properties of Logarithms
- and

### Practice

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

Find the inverse of each of the following exponential functions.