3.3.1: Product and Quotient Properties of Logarithms

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 log₃162−log₃2. "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?

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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_bx + log_by.

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

log_bx = m and log_by = n, then b^m = x and bⁿ=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 log₁₂4y.

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

log₁₂4y = log₁₂4 + log₁₂y

Finally, let's simplify log₃15−log₃5.

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:

\(\ \begin{aligned}
\log _{3} 15-\log _{3} 5 &=\log _{3} \frac{15}{5} \\
&=\log _{3} 3 \\
&=1
\end{aligned}\)

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Examples

Example 1

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

Solution

If you rewrite log₃162−log₃2 as log₃\(\ \frac{162}{2}\), you get log₃81.

3⁴=81 so you each owe $4.

Example 2

Simplify the following expression: log₇8 + log₇x² + log₇3y.

Solution

Combine all the logs together using the Product Property.

log₇8 + log₇x² + log₇3y = log₇8x²3y

=log₇24x²y

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: log₂32 − log₂z.

Solution

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

log₂32−log₂z=5−log₂z

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 log₈16, use the definition and powers of 2:

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

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Review

Simplify the following logarithmic expressions.

1. log₃ 6 + log₃ y − log₃ 4
2. log12 − logx+log y²
3. log₆ x² − log₆x − log₆y
4. ln8 + ln6 − ln12
5. ln7 − ln14 + ln10
6. log₁₁ 22 + log₁₁ 5 − log₁₁ 55

Expand the following logarithmic functions.

7. log₆ (5x)
8. log₃ (abc)
9. \(\ \log \left(\frac{a^{2}}{b}\right)\)
10. \(\ \log _{9}\left(\frac{x y}{5}\right)\)
11. \(\ \log \left(\frac{2 x}{y}\right)\)
12. \(\ \log \left(\frac{8 x^{2}}{15}\right)\)
13. \(\ \log _{4}\left(\frac{5}{9 y}\right)\)
14. 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.

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Answers for Review Problems

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

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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 \(\ \log _{b} \frac{x}{y}=\log _{b} x-\log _{b} y\).