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# Product and Quotient Properties of Logarithms

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Product and Quotient Properties of Logarithms

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 $\log_3 162 - \log_3 2$ . "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 $\log_b x + \log_b y$ .

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

$\log_b x=m$ and $\log_b y=n$ , then $b^m=x$ and $b^n=y$ .

Now, multiply the latter two equations together.

$b^m \cdot b^n &= xy \\b^{m+n} &= xy$

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

$b^{m+n}=xy \rightarrow \log_b xy=m+n$

Recall that $m=\log_b x$ and $n=\log_b y$ , therefore $\log_b xy=\log_b x + \log_b y$ .

This is the Product Property of Logarithms .

#### Example B

Expand $\log_{12} 4y$ .

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

$\log_{12} 4y = \log_{12} 4 + \log_{12} y$

#### Example C

Simplify $\log_3 15 - \log_3 5$ .

Solution: As you might expect, the Quotient Property of Logarithms is $\log_b \frac{x}{y}=\log_b x - \log_b y$ (proof in the Problem Set). Therefore, the answer is:

$\log_3 15 - \log_3 5 &= \log_3 \frac{15}{5} \\&= \log_3 3 \\&= 1$

Intro Problem Revisit

If you rewrite $\log_3 162 - \log_3 2$ as $\log_3 \frac {162}{2}$ , you get $\log_3 81$ .

$3^4 = 81$ so you each owe \$4.

### Guided Practice

Simplify the following expressions.

1. $\log_7 8 + \log_7 x^2 + \log_7 3y$

2. $\log y - \log 20 + \log 8x$

3. $\log_2 32 - \log_2 z$

4. $\log_8 \frac{16x}{y^2}$

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

$\log_7 8 + \log_7 x^2 + \log_7 3y &= \log_7 8x^2 3y \\&= \log_7 24x^2 y$

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

$\log y - \log 20 + \log 8x &= \log \frac{y}{20} \cdot 8x \\&= \log \frac{2xy}{5}$

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

$\log_2 32 - \log_2 z=5 - \log_2 z$

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

$\log_8 \frac{16x}{y^2} &= \log_8 16x - \log_8 y^2 \\&= \log_8 16 + \log_8 x-\log_8 y^2 \\&= \frac{4}{3} + \log_8 x - \log_8 y^2$

To determine $\log_8 16$ , use the definition and powers of 2: $8^n=16 \rightarrow 2^{3n}=2^4 \rightarrow 3n = 4 \rightarrow n=\frac{4}{3}$ .

### Vocabulary

Product Property of Logarithms
As long as $b \ne 1$ , then $\log_b xy=\log_b x + \log_b y$
Quotient Property of Logarithms
As long as $b \ne 1$ , then $\log_b \frac{x}{y}=\log_b x - \log_b y$

### Explore More

Simplify the following logarithmic expressions.

1. $\log_3 6 + \log_3 y - \log_3 4$
2. $\log 12 - \log x + \log y^2$
3. $\log_6 x^2 - \log_6 x - \log_6 y$
4. $\ln 8 + \ln 6 - \ln 12$
5. $\ln 7 - \ln 14 + \ln 10$
6. $\log_{11} 22 + \log_{11} 5 - \log_{11} 55$

Expand the following logarithmic functions.

1. $\log_6 (5x)$
2. $\log_3 (abc)$
3. $\log \left(\frac{a^2}{b}\right)$
4. $\log_9 \left(\frac{xy}{5}\right)$
5. $\log \left(\frac{2x}{y}\right)$
6. $\log \left(\frac{8x^2}{15}\right)$
7. $\log_4 \left(\frac{5}{9y}\right)$
8. Write an algebraic proof of the Quotient Property. Start with the expression $\log_a x - \log_a y$ and the equations $\log_a x=m$ and $\log_a y=n$ in your proof. Refer to the proof of the product property in Example A as a guide for your proof.