# 3.3.2: Power Property of Logarithms


## Power Property of Logarithms

The hypotenuse of a right triangle has a length of log3 278. How long is the triangle's hypotenuse?

## Power Property

The last property of logs is the Power Property.

logbx=y

Using the definition of a log, we have by=x. Now, raise both sides to the n power.

\ \begin{aligned} \left(b^{y}\right)^{n} &=x^{n} \\ b^{n y} &=x^{n} \end{aligned}

Let’s convert this back to a log with base b, logb xn = ny. Substituting for y, we have logb xn = n logb x.

Therefore, the Power Property says that if there is an exponent within a logarithm, we can pull it out in front of the logarithm.

Let's use the Power Property to expand the following logarithms.

1. To expand this log, we need to use the Product Property and the Power Property.

\ \begin{aligned} \log _{6} 17 x^{5} &=\log _{6} 17+\log _{6} x^{5} \\ &=\log _{6} 17+5 \log _{6} x \end{aligned}

2. $$\ \ln \left(\frac{2 x}{y^{3}}\right)^{4}$$ We will need to use all three properties to expand this problem. Because the expression within the natural log is in parenthesis, start with moving the 4th power to the front of the log.

\ \begin{aligned} \ln \left(\frac{2 x}{y^{3}}\right)^{4} &=4 \ln \frac{2 x}{y^{3}} \\ &=4\left(\ln 2 x-\ln y^{3}\right) \\ &=4(\ln 2+\ln x-3 \ln y) \\ &=4 \ln 2+4 \ln x-12 \ln y \end{aligned}

Depending on how your teacher would like your answer, you can evaluate 4 ln2 ≈ 2.77, making the final answer 2.77 + 4 lnx − 12 ln y.

Now, let's condense log 9 − 4 log 5 − 4 log x + 2 log 7 + 2 log y.

This is the opposite of the previous two problems. Start with the Power Property.

log 9 − 4 log 5 − 4 log x + 2 log 7 + 2 log y

log 9 − log 54 − log x4 + log 72 + log y2

Now, start changing things to division and multiplication within one log.

$$\ \log \frac{9 \cdot 7^{2} y^{2}}{5^{4} x^{4}}$$

Lastly, combine like terms.

$$\ \log \frac{441 y^{2}}{625 x^{4}}$$

## Examples

###### Example 1

Earlier, you were asked to find the length of the triangle's hypotenuse.

Solution

We can rewrite log3 278 and 8 log3 27 and solve.

8 log3 27

=8⋅3

=24

Therefore, the triangle's hypotenuse is 24 units long.

###### Example 2

Expand the following expression: ln x3.

Solution

The only thing to do here is apply the Power Property: 3 lnx.

###### Example 3

Expand the following expression: $$\ \log _{16} \frac{x^{2} y}{32 z^{5}}$$.

Solution

$$\ \log _{16} \frac{x^{2} y}{32 z^{5}}=\log _{16} x^{2} y-\log _{16} 32 z^{5}$$

Now, apply the Product Property, followed by the Power Property.

$$\ \begin{array}{l} =\log _{16} x^{2}+\log _{16} y-\left(\log _{16} 32+\log _{16} z^{5}\right) \\ =2 \log _{16} x+\log _{16} y-\frac{5}{4}-5 \log _{16} z \end{array}$$

Simplify $$\ \log _{16} 32 \rightarrow 16^{n}=32 \rightarrow 2^{4 n}=2^{5}$$ and solve for $$\ n$$. Also, notice that we put parenthesis around the second log once it was expanded to ensure that the $$\ z^5$$ would also be subtracted (because it was in the denominator of the original expression).

###### Example 4

Expand the following expression: $$\ \log \left(5 c^{4}\right)^{2}$$.

Solution

For this problem, you will need to apply the Power Property twice.

\ \begin{aligned} \log \left(5 c^{4}\right)^{2} &=2 \log 5 c^{4} \\ &=2\left(\log 5+\log c^{4}\right) \\ &=2(\log 5+4 \log c) \\ &=2 \log 5+8 \log c \end{aligned}

Important Note: You can write this particular log several different ways. Equivalent logs are: $$\ \log 25+8 \log c, \log 25+\log c^{8} \text { and } \log 25 c^{8}$$. Because of these properties, there are several different ways to write one logarithm.

###### Example 5

Condense into one log: $$\ \ln 5-7 \ln x^{4}+2 \ln y$$.

Solution

To condense this expression into one log, you will need to use all three properties.

\ \begin{aligned} \ln 5-7 \ln x^{4}+2 \ln y &=\ln 5-\ln x^{28}+\ln y^{2} \\ &=\ln \frac{5 y^{2}}{x^{28}} \end{aligned}

Important Note: If the problem was $$\ \ln 5-\left(7 \ln x^{4}+2 \ln y\right)$$, then the answer would have been $$\ \ln \frac{5}{x^{28} y^{2}}$$. But, because there are no parentheses, the $$\ y^2$$ is in the numerator.

## Review

Expand the following logarithmic expressions.

1. $$\ \log _{7} y^{2}$$
2. $$\ \log _{12} 5 z^{2}$$
3. $$\ \log _{4}(9 x)^{3}$$
4. $$\ \log \left(\frac{3 x}{y}\right)^{2}$$
5. $$\ \log _{8} \frac{x^{3} y^{2}}{z^{4}}$$
6. $$\ \log _{5}\left(\frac{25 x^{4}}{y}\right)^{2}$$
7. $$\ \ln \left(\frac{6 x}{y^{3}}\right)^{-2}$$
8. $$\ \ln \left(\frac{e^{5} x^{-2}}{y^{3}}\right)^{6}$$

Condense the following logarithmic expressions.

1. $$\ 6 \log x$$
2. $$\ 2 \log _{6} x+5 \log _{6} y$$
3. $$\ 3(\log x-\log y)$$
4. $$\ \frac{1}{2} \log (x+1)-3 \log y$$
5. $$\ 4 \log _{2} y+\frac{1}{3} \log _{2} x^{3}$$
6. $$\ \frac{1}{5}\left[10 \log _{2}(x-3)+\log _{2} 32-\log _{2} y\right]$$
7. $$\ 4\left[\frac{1}{2} \log _{3} y-\frac{1}{3} \log _{3} x-\log _{3} z\right]$$

Power Property The power property for logarithms states that as long as $$\ b≠1$$, then $$\ \log _{b} x^{n}=n \log _{b} x$$.