# 3.3.1: 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?

# 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 log_{b}x + log_{b}y.

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.

\(\ \begin{aligned}

b^{m} \cdot b^{n} &=x y \\

b^{m+n} &=x y

\end{aligned}\)

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

\(\ b^{m+n}=x y \rightarrow \log _{b} x y=m+n\)

Recall that \(\ m=\log _{b} x \text { and } n=\log _{b} y, \text { therefore } \log _{b} x y=\log _{b} x+\log _{b} y\).

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

Now, let's expand log_{12}4y.

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

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

Finally, let's simplify log_{3}15−log_{3}5.

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

\(\ \begin{aligned}

\log _{3} 15-\log _{3} 5 &=\log _{3} \frac{15}{5} \\

&=\log _{3} 3 \\

&=1

\end{aligned}\)

# Examples

Example 1

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

**Solution**

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.

Example 2

Simplify the following expression: log_{7}8 + log_{7}x^{2 }+ log_{7}3y.

**Solution**

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

Example 3

Simplify the following expression: log y−log 20+log 8x.

**Solution**

Use both the Product and Quotient Property to condense.

\(\ \begin{aligned}

\log y-\log 20+\log 8 x &=\log \frac{y}{20} \cdot 8 x \\

&=\log \frac{2 x y}{5}

\end{aligned}\)

Example 4

Simplify the following expression: log_{2}32 − log_{2}z.

**Solution**

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

Example 5

Simplify the following expression: \(\ \log _{8} \frac{16 x}{y^{2}}\).

**Solution**

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

\(\ \begin{aligned}

\log _{8} \frac{16 x}{y^{2}} &=\log _{8} 16 x-\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{aligned}\)

To determine log_{8}16, use the definition and powers of 2:

\(\ 8^{n}=16 \rightarrow 2^{3 n}=2^{4} \rightarrow 3 n=4 \rightarrow n=\frac{4}{3}\)

# Review

Simplify the following logarithmic expressions.

- log
_{3 }6 + log_{3 }y − log_{3 }4 - log12 − logx+log y
^{2} - log
_{6 }x^{2 }− log_{6}x − log_{6}y - ln8 + ln6 − ln12
- ln7 − ln14 + ln10
- log
_{11 }22 + log_{11 }5 − log_{11 }55

Expand the following logarithmic functions.

- log
_{6 }(5x) - log
_{3 }(abc) - \(\ \log \left(\frac{a^{2}}{b}\right)\)
- \(\ \log _{9}\left(\frac{x y}{5}\right)\)
- \(\ \log \left(\frac{2 x}{y}\right)\)
- \(\ \log \left(\frac{8 x^{2}}{15}\right)\)
- \(\ \log _{4}\left(\frac{5}{9 y}\right)\)
- 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 the first practice problem as a guide for your proof.

# Vocabulary

Term | Definition |
---|---|

Product Property of Logarithms |
The product property of logarithms states that as long as \(\ b≠1\), then \(\ \log _{b} x y=\log _{b} x+\log _{b} y\) |

Quotient Property of Logarithms |
The quotient property of logarithms states that as long as \(\ b≠1\), then \(\ \log _{b} \frac{x}{y}=\log _{b} x-\log _{b} y\). |