# 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?

# 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 logbx + logby.

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

logbx = m and logby = n, then b= 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}

# 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 + log7x+ log73y.

Solution

Combine all the logs together using the Product Property.

log78 + log7x+ 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:

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

# Review

Simplify the following logarithmic expressions.

1. log6 + logy − log4
2. log12 − logx+log y2
3. logx− log6x − log6y
4. ln8 + ln6 − ln12
5. ln7 − ln14 + ln10
6. log11 22 + log11 5 − log11 55

Expand the following logarithmic functions.

1. log(5x)
2. log(abc)
3. $$\ \log \left(\frac{a^{2}}{b}\right)$$
4. $$\ \log _{9}\left(\frac{x y}{5}\right)$$
5. $$\ \log \left(\frac{2 x}{y}\right)$$
6. $$\ \log \left(\frac{8 x^{2}}{15}\right)$$
7. $$\ \log _{4}\left(\frac{5}{9 y}\right)$$
8. Write an algebraic proof of the Quotient Property. Start with the expression logx − logy and the equations logx = m and logy = n in your proof. Refer to the proof of the Product Property in the first practice problem as a guide for your proof.

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