# 3.3.2: Power Property of Logarithms

- Page ID
- 14373

## Power Property of Logarithms

The hypotenuse of a right triangle has a length of log_{3 }27^{8}. How long is the triangle's hypotenuse?

## Power Property

The last property of logs is the **Power Property**.

log_{b}x=y

Using the definition of a log, we have b^{y}=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, log_{b }x^{n }= ny. Substituting for y, we have log_{b }x^{n }= n log_{b }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.

- 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}\) - \(\ \ln \left(\frac{2 x}{y^{3}}\right)^{4}\)
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 4
^{th}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 5

^{4 }− log x^{4 }+ log 7^{2 }+ log y^{2}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

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

**Solution**

We can rewrite log_{3 }27^{8} and 8 log_{3 }27 and solve.

8 log_{3 }27

=8⋅3

=24

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

Expand the following expression: ln x^{3}.

**Solution**

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

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

**Solution**

Let’s start with using the Quotient Property.

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

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).

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.

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.

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

Condense the following logarithmic expressions.

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

## Vocabulary

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

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