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8.9: Power Property of Logarithms

Difficulty Level: At Grade Created by: CK-12
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The hypotenuse of a right triangle has a length of log3278. How long is the triangle's hypotenuse?

Guidance

The last property of logs is the Power Property.

logbx=y

Using the definition of a log, we have by=x. Now, raise both sides to the n power.

(by)nbny=xn=xn

Let’s convert this back to a log with base b, logbxn=ny. Substituting for y, we have logbxn=nlogbx.

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.

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

log617x5=log617+log6x5=log617+5log6x

Example B

Expand ln(2xy3)4.

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 power to the front of the log.

ln(2xy3)4=4ln2xy3=4(ln2xlny3)=4(ln2+lnx3lny)=4ln2+4lnx12lny

Depending on how your teacher would like your answer, you can evaluate 4ln22.77, making the final answer 2.77+4lnx12lny.

Example C

Condense log94log54logx+2log7+2logy.

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

log94log54logx+2log7+2logylog9log54logx4+log72+logy2

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

log972y254x4

Lastly, combine like terms.

log441y2625x4

Intro Problem Revisit We can rewrite log3278 and 8log327 and solve.

8log327=83=24

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

Guided Practice

Expand the following logarithmic expressions.

1. lnx3

2. log16x2y32z5

3. log(5c4)2

4. Condense into one log: ln57lnx4+2lny.

Answers

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

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

log16x2y32z5=log16x2ylog1632z5

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

=log16x2+log16y(log1632+log16z5)=2log16x+log16y545log16z

Simplify log163216n=3224n=25 and solve for n. Also, notice that we put parenthesis around the second log once it was expanded to ensure that the z5 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

Important Note: You can write this particular log several different ways. Equivalent logs are: log25+8logc,log25+logc8 and log25c8. 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

Important Note: If the problem was ln5(7lnx4+2lny), then the answer would have been ln5x28y2. But, because there are no parentheses, the y2 is in the numerator.

Vocabulary

Power Property
As long as b1, then logbxn=nlogbx.

Practice

Expand the following logarithmic expressions.

  1. log7y2
  2. log125z2
  3. log4(9x)3
  4. log(3xy)2
  5. log8x3y2z4
  6. log5(25x4y)2
  7. ln(6xy3)2
  8. ln(e5x2y3)6

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

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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