Your friend Robbie works as a server at a pizza parlor. You and two of your friends go to the restaurant and order a pizza. You ask Robbie to bring you separate checks so you can split the cost of the pizza. Instead of bringing you three checks, Robbie brings you one with the total . "This is how much each of you owes," he says as he drops the bill on the table. How much do each of you owe?

### Guidance

Just like exponents, logarithms have special properties, or shortcuts, that can be applied when simplifying expressions. In this lesson, we will address two of these properties.

#### Example A

Simplify .

**
Solution:
**
First, notice that these logs have the same base. If they do not, then the properties do not apply.

and , then and .

Now, multiply the latter two equations together.

Recall, that when two exponents with the same base are multiplied, we can add the exponents. Now, reapply the logarithm to this equation.

Recall that and , therefore .

This is the
**
Product Property of Logarithms
**
.

#### Example B

Expand .

**
Solution:
**
Applying the Product Property from Example A, we have:

#### Example C

Simplify .

**
Solution:
**
As you might expect, the
**
Quotient Property of Logarithms
**
is
(proof in the Problem Set). Therefore, the answer is:

**
Intro Problem Revisit
**

If you rewrite as , you get .

so you each owe $4.

### Guided Practice

Simplify the following expressions.

1.

2.

3.

4.

#### Answers

1. Combine all the logs together using the Product Property.

2. Use both the Product and Quotient Property to condense.

3. Be careful; you do not have to use either rule here, just the definition of a logarithm.

4. When expanding a log, do the division first and then break the numerator apart further.

To determine , use the definition and powers of 2: .

### Vocabulary

- Product Property of Logarithms
- As long as , then

- Quotient Property of Logarithms
- As long as , then

### Practice

Simplify the following logarithmic expressions.

Expand the following logarithmic functions.

- Write an algebraic proof of the Quotient Property. Start with the expression and the equations and in your proof. Refer to the proof of the product property in Example A as a guide for your proof.