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# Product and Quotient Properties of Logarithms

## Combine and expand logarithmic expressions

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Practice 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 \begin{align*}\log_3 162 - \log_3 2\end{align*}. "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?

### Product and Quotient Properties of Logarithms

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.

Let's simplify \begin{align*}\log_b x + \log_b y\end{align*}.

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

\begin{align*}\log_b x=m\end{align*} and \begin{align*}\log_b y=n\end{align*}, then \begin{align*}b^m=x\end{align*} and \begin{align*}b^n=y\end{align*}.

Now, multiply the latter two equations together.

\begin{align*}b^m \cdot b^n &= xy \\ b^{m+n} &= xy\end{align*}

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

\begin{align*}b^{m+n}=xy \rightarrow \log_b xy=m+n\end{align*}

Recall that \begin{align*}m=\log_b x\end{align*} and \begin{align*}n=\log_b y\end{align*}, therefore \begin{align*}\log_b xy=\log_b x + \log_b y\end{align*}.

This is the Product Property of Logarithms.

Now, let's expand \begin{align*}\log_{12} 4y\end{align*}.

Applying the Product Property from the previous problem, we have:

\begin{align*}\log_{12} 4y = \log_{12} 4 + \log_{12} y\end{align*}

Finally, let's simplify \begin{align*}\log_3 15 - \log_3 5\end{align*}.

As you might expect, the Quotient Property of Logarithms is \begin{align*}\log_b \frac{x}{y}=\log_b x - \log_b y\end{align*} (proof in the Review section). Therefore, the answer is:

\begin{align*}\log_3 15 - \log_3 5 &= \log_3 \frac{15}{5} \\ &= \log_3 3 \\ &= 1\end{align*}

### Examples

#### Example 1

Earlier, you were asked to find the amount that each of you owes.

If you rewrite \begin{align*}\log_3 162 - \log_3 2\end{align*} as \begin{align*}\log_3 \frac {162}{2}\end{align*}, you get \begin{align*}\log_3 81\end{align*}.

\begin{align*}3^4 = 81\end{align*} so you each owe \$4.

#### Example 2

Simplify the following expression: \begin{align*}\log_7 8 + \log_7 x^2 + \log_7 3y\end{align*}.

Combine all the logs together using the Product Property.

\begin{align*}\log_7 8 + \log_7 x^2 + \log_7 3y &= \log_7 8x^2 3y \\ &= \log_7 24x^2 y\end{align*}

#### Example 3

Simplify the following expression: \begin{align*}\log y - \log 20 + \log 8x\end{align*}.

Use both the Product and Quotient Property to condense.

\begin{align*}\log y - \log 20 + \log 8x &= \log \frac{y}{20} \cdot 8x \\ &= \log \frac{2xy}{5}\end{align*}

#### Example 4

Simplify the following expression: \begin{align*}\log_2 32 - \log_2 z\end{align*}.

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

\begin{align*}\log_2 32 - \log_2 z=5 - \log_2 z\end{align*}

#### Example 5

Simplify the following expression: \begin{align*}\log_8 \frac{16x}{y^2}\end{align*}.

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

\begin{align*}\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\end{align*}

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

### Review

Simplify the following logarithmic expressions.

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

Expand the following logarithmic functions.

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

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 8.8.

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### Vocabulary Language: English

TermDefinition
Product Property of Logarithms The product property of logarithms states that as long as $b \ne 1$, then $\log_b xy=\log_b x + \log_b y$
Quotient Property of Logarithms The quotient property of logarithms states that as long as $b \ne 1$, then $\log_b \frac{x}{y}=\log_b x - \log_b y$.