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Power Property of Logarithms

Manipulate exponents in logarithms

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Power Property of Logarithms

The hypotenuse of a right triangle has a length of \begin{align*}\log_3 27^8\end{align*}. How long is the triangle's hypotenuse?

Power Property

The last property of logs is the Power Property.

\begin{align*}\log_b x=y\end{align*}

Using the definition of a log, we have \begin{align*}b^y=x\end{align*}. Now, raise both sides to the \begin{align*}n\end{align*} power.

\begin{align*}(b^y)^n &= x^n \\ b^{ny} &= x^n\end{align*}

Let’s convert this back to a log with base \begin{align*}b\end{align*}, \begin{align*}\log_b x^n=ny\end{align*}. Substituting for \begin{align*}y\end{align*}, we have \begin{align*}\log_b x^n=n \log_b x\end{align*}.

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

Use the Power Property to solve the following problems

Expand \begin{align*}\log_6 17x^5\end{align*}.

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

\begin{align*}\log_6 17x^5 &= \log_6 17 + \log_6 x^5 \\ &= \log_6 17 + 5\log_6 x\end{align*}

Expand \begin{align*}\ln \left(\frac{2x}{y^3}\right)^4\end{align*}.

We will need to use all three properties to expand this example. Because the expression within the natural log is in parenthesis, start with moving the \begin{align*}4^{th}\end{align*} power to the front of the log.

\begin{align*}\ln \left(\frac{2x}{y^3}\right)^4 &= 4 \ln \frac{2x}{y^3} \\ &= 4(\ln 2x - \ln y^3)\\ &= 4(\ln 2 + \ln x - 3 \ln y) \\ &= 4 \ln2 + 4 \ln x - 12 \ln y\end{align*}

Depending on how your teacher would like your answer, you can evaluate \begin{align*}4\ln2 \approx 2.77\end{align*}, making the final answer \begin{align*}2.77 + 4\ln x - 12\ln y\end{align*}.

Condense \begin{align*}\log 9 - 4\log 5 - 4\log x + 2\log 7 + 2\log y\end{align*}.

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

\begin{align*}&\log 9 - 4\log 5 - 4\log x + 2\log7 + 2\log y \\ &\log 9 - \log 5^4 - \log x^4 + \log 7^2 + \log y^2\end{align*}

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

\begin{align*}\log \frac{9 \cdot 7^2 y^2}{5^4 x^4}\end{align*}

Lastly, combine like terms.

\begin{align*}\log \frac{441 y^2}{625 x^4}\end{align*}

Examples

Example 1

Earlier, you were asked how long is the triangle's hypotenuse. 

We can rewrite \begin{align*}\log_3 27^8\end{align*} and \begin{align*}8\log_3 27\end{align*} and solve.

\begin{align*}8\log_3 27\\ = 8 \cdot 3\\ = 24\end{align*}

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

Expand the following logarithmic expressions.

Example 2

\begin{align*}\ln x^3\end{align*}

The only thing to do here is apply the Power Property: \begin{align*}3 \ln x\end{align*}.

Example 3

\begin{align*}\log_{16} \frac{x^2 y}{32 z^5}\end{align*}

Let’s start with using the Quotient Property.

\begin{align*}\log_{16} \frac{x^2 y}{32 z^5} = \log_{16} x^2y - \log_{16} 32z^5\end{align*}

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

\begin{align*}&= \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{align*}

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

Example 4

\begin{align*}\log (5c^4)^2\end{align*}

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

\begin{align*}\log (5c^4)^2 &= 2 \log 5c^4 \\ &= 2(\log 5 + \log c^4) \\ &= 2(\log 5 + 4 \log c) \\ &= 2 \log 5 + 8 \log c\end{align*}

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

Example 5

Condense into one log: \begin{align*}\ln 5 - 7 \ln x^4 + 2 \ln y\end{align*}.

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

\begin{align*}\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{align*}

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

Review

Expand the following logarithmic expressions.

  1. \begin{align*}\log_7 y^2\end{align*}
  2. \begin{align*}\log_{12} 5z^2\end{align*}
  3. \begin{align*}\log_4 (9x)^3\end{align*}
  4. \begin{align*}\log \left(\frac{3x}{y}\right)^2\end{align*}
  5. \begin{align*}\log_8 \frac{x^3 y^2}{z^4}\end{align*}
  6. \begin{align*}\log_5 \left(\frac{25x^4}{y}\right)^2\end{align*}
  7. \begin{align*}\ln \left(\frac{6x}{y^3}\right)^{-2}\end{align*}
  8. \begin{align*}\ln \left(\frac{e^5 x^{-2}}{y^3}\right)^6\end{align*}

Condense the following logarithmic expressions.

  1. \begin{align*}6 \log x\end{align*}
  2. \begin{align*}2 \log_6 x + 5 \log_6 y\end{align*}
  3. \begin{align*}3(\log x - \log y)\end{align*}
  4. \begin{align*}\frac{1}{2} \log(x+1) - 3 \log y\end{align*}
  5. \begin{align*}4 \log_2 y + \frac{1}{3} \log_2 x^3\end{align*}
  6. \begin{align*}\frac{1}{5} \left[10 \log_2 (x-3) + \log_2 32 - \log_2 y \right]\end{align*}
  7. \begin{align*}4 \left[\frac{1}{2} \log_3 y - \frac{1}{3} \log_3 x - \log_3 z \right]\end{align*}

Answers for Review Problems

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

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Vocabulary

Power Property

The power property for logarithms states that as long as b \ne 1, then \log_b x^n = n \log_b x.

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