8.9: Power Property of Logarithms
The hypotenuse of a right triangle has a length of
Guidance
The last property of logs is the Power Property.
Using the definition of a log, we have
Let’s convert this back to a log with base
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
Solution: To expand this log, we need to use the Product Property and the Power Property.
Example B
Expand
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
Depending on how your teacher would like your answer, you can evaluate
Example C
Condense
Solution: This is the opposite of the previous two examples. Start with the Power Property.
Now, start changing things to division and multiplication within one log.
Lastly, combine like terms.
Intro Problem Revisit We can rewrite
Therefore, the triangle's hypotenuse is 24 units long.
Guided Practice
Expand the following logarithmic expressions.
1.
2.
3.
4. Condense into one log:
Answers
1. The only thing to do here is apply the Power Property:
2. Let’s start with using the Quotient Property.
Now, apply the Product Property, followed by the Power Property.
Simplify
3. For this problem, you will need to apply the Power Property twice.
Important Note: You can write this particular log several different ways. Equivalent logs are:
4. To condense this expression into one log, you will need to use all three properties.
Important Note: If the problem was
Vocabulary
 Power Property

As long as
b≠1 , thenlogbxn=nlogbx .
Practice
Expand the following logarithmic expressions.

log7y2 
log125z2 
log4(9x)3 
log(3xy)2 
log8x3y2z4 
log5(25x4y)2 
ln(6xy3)−2 
ln(e5x−2y3)6
Condense the following logarithmic expressions.
 \begin{align*}6 \log x\end{align*}
 \begin{align*}2 \log_6 x + 5 \log_6 y\end{align*}
 \begin{align*}3(\log x  \log y)\end{align*}
 \begin{align*}\frac{1}{2} \log(x+1)  3 \log y\end{align*}
 \begin{align*}4 \log_2 y + \frac{1}{3} \log_2 x^3\end{align*}
 \begin{align*}\frac{1}{5} \left[10 \log_2 (x3) + \log_2 32  \log_2 y \right]\end{align*}
 \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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Here you'll use the Power Property of logarithms.