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# 8.9: Power Property of Logarithms

Difficulty Level: At Grade Created by: CK-12
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Practice Power Property of Logarithms
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The hypotenuse of a right triangle has a length of log3278\begin{align*}\log_3 27^8\end{align*}. How long is the triangle's hypotenuse?

### Guidance

The last property of logs is the Power Property.

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

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

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

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

#### Example A

Expand log617x5\begin{align*}\log_6 17x^5\end{align*}.

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

log617x5=log617+log6x5=log617+5log6x\begin{align*}\log_6 17x^5 &= \log_6 17 + \log_6 x^5 \\ &= \log_6 17 + 5\log_6 x\end{align*}

#### Example B

Expand ln(2xy3)4\begin{align*}\ln \left(\frac{2x}{y^3}\right)^4\end{align*}.

Solution: 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 4th\begin{align*}4^{th}\end{align*} power to the front of the log.

ln(2xy3)4=4ln2xy3=4(ln2xlny3)=4(ln2+lnx3lny)=4ln2+4lnx12lny\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 4ln22.77\begin{align*}4\ln2 \approx 2.77\end{align*}, making the final answer 2.77+4lnx12lny\begin{align*}2.77 + 4\ln x - 12\ln y\end{align*}.

#### Example C

Condense log94log54logx+2log7+2logy\begin{align*}\log 9 - 4\log 5 - 4\log x + 2\log 7 + 2\log y\end{align*}.

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

log94log54logx+2log7+2logylog9log54logx4+log72+logy2\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.

log972y254x4\begin{align*}\log \frac{9 \cdot 7^2 y^2}{5^4 x^4}\end{align*}

Lastly, combine like terms.

log441y2625x4\begin{align*}\log \frac{441 y^2}{625 x^4}\end{align*}

Intro Problem Revisit We can rewrite log3278\begin{align*}\log_3 27^8\end{align*} and 8log327\begin{align*}8\log_3 27\end{align*} and solve.

8log327=83=24\begin{align*}8\log_3 27\\ = 8 \cdot 3\\ = 24\end{align*}

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

### Guided Practice

Expand the following logarithmic expressions.

1. lnx3\begin{align*}\ln x^3\end{align*}

2. log16x2y32z5\begin{align*}\log_{16} \frac{x^2 y}{32 z^5}\end{align*}

3. log(5c4)2\begin{align*}\log (5c^4)^2\end{align*}

4. Condense into one log: ln57lnx4+2lny\begin{align*}\ln 5 - 7 \ln x^4 + 2 \ln y\end{align*}.

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

2. Let’s start with using the Quotient Property.

log16x2y32z5=log16x2ylog1632z5\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.

=log16x2+log16y(log1632+log16z5)=2log16x+log16y545log16z\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 log163216n=3224n=25\begin{align*}\log_{16} 32 \rightarrow 16^n = 32 \rightarrow 2^{4n} = 2^5\end{align*} and solve for n\begin{align*}n\end{align*}. Also, notice that we put parenthesis around the second log once it was expanded to ensure that the z5\begin{align*}z^5\end{align*} would also be subtracted (because it was in the denominator of the original expression).

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

log(5c4)2=2log5c4=2(log5+logc4)=2(log5+4logc)=2log5+8logc\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: log25+8logc,log25+logc8\begin{align*}\log 25 + 8 \log c, \log 25 + \log c^8\end{align*} and log25c8\begin{align*}\log 25c^8\end{align*}. Because of these properties, there are several different ways to write one logarithm.

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

ln57lnx4+2lny=ln5lnx28+lny2=ln5y2x28\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 ln5(7lnx4+2lny)\begin{align*}\ln 5 - (7 \ln x^4 + 2 \ln y)\end{align*}, then the answer would have been ln5x28y2\begin{align*}\ln \frac{5}{x^{28}y^2}\end{align*}. But, because there are no parentheses, the y2\begin{align*}y^2\end{align*} is in the numerator.

### Vocabulary

Power Property
As long as b1\begin{align*}b \ne 1\end{align*}, then logbxn=nlogbx\begin{align*}\log_b x^n = n \log_b x\end{align*}.

### Practice

Expand the following logarithmic expressions.

1. log7y2\begin{align*}\log_7 y^2\end{align*}
2. log125z2\begin{align*}\log_{12} 5z^2\end{align*}
3. log4(9x)3\begin{align*}\log_4 (9x)^3\end{align*}
4. log(3xy)2\begin{align*}\log \left(\frac{3x}{y}\right)^2\end{align*}
5. log8x3y2z4\begin{align*}\log_8 \frac{x^3 y^2}{z^4}\end{align*}
6. log5(25x4y)2\begin{align*}\log_5 \left(\frac{25x^4}{y}\right)^2\end{align*}
7. ln(6xy3)2\begin{align*}\ln \left(\frac{6x}{y^3}\right)^{-2}\end{align*}
8. ln(e5x2y3)6\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*}

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### Vocabulary Language: English

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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