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?

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Power Property

The last property of logs is the Power Property.

log_bx=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 xⁿ = ny. Substituting for y, we have log_b xⁿ = 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.

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 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 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.

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

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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

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).

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.

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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.

9. \(\ 6 \log x\)
10. \(\ 2 \log _{6} x+5 \log _{6} y\)
11. \(\ 3(\log x-\log y)\)
12. \(\ \frac{1}{2} \log (x+1)-3 \log y\)
13. \(\ 4 \log _{2} y+\frac{1}{3} \log _{2} x^{3}\)
14. \(\ \frac{1}{5}\left[10 \log _{2}(x-3)+\log _{2} 32-\log _{2} y\right]\)
15. \(\ 4\left[\frac{1}{2} \log _{3} y-\frac{1}{3} \log _{3} x-\log _{3} z\right]\)

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Answers for Review Problems

To see the Review answers, open this PDF file and look for section 8.9.

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Vocabulary

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