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# Power Property of Logarithms

## Manipulate exponents in logarithms

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Practice Power Property of Logarithms
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Power Property of Logarithms

The hypotenuse of a right triangle has a length of $\log_3 27^8$ . How long is the triangle's hypotenuse?

### Guidance

The last property of logs is the Power Property .

$\log_b x=y$

Using the definition of a log, we have $b^y=x$ . Now, raise both sides to the $n$ power.

$(b^y)^n &= x^n \\b^{ny} &= x^n$

Let’s convert this back to a log with base $b$ , $\log_b x^n=ny$ . Substituting for $y$ , we have $\log_b x^n=n \log_b x$ .

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

#### Example A

Expand $\log_6 17x^5$ .

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

$\log_6 17x^5 &= \log_6 17 + \log_6 x^5 \\&= \log_6 17 + 5\log_6 x$

#### Example B

Expand $\ln \left(\frac{2x}{y^3}\right)^4$ .

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 $4^{th}$ power to the front of the log.

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

Depending on how your teacher would like your answer, you can evaluate $4\ln2 \approx 2.77$ , making the final answer $2.77 + 4\ln x - 12\ln y$ .

#### Example C

Condense $\log 9 - 4\log 5 - 4\log x + 2\log 7 + 2\log y$ .

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

$&\log 9 - 4\log 5 - 4\log x + 2\log7 + 2\log y \\&\log 9 - \log 5^4 - \log x^4 + \log 7^2 + \log y^2$

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.

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

Intro Problem Revisit We can rewrite $\log_3 27^8$ and $8\log_3 27$ and solve.

$8\log_3 27\\= 8 \cdot 3\\= 24$

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

### Guided Practice

Expand the following logarithmic expressions.

1. $\ln x^3$

2. $\log_{16} \frac{x^2 y}{32 z^5}$

3. $\log (5c^4)^2$

4. Condense into one log: $\ln 5 - 7 \ln x^4 + 2 \ln y$ .

1. The only thing to do here is apply the Power Property: $3 \ln x$ .

$\log_{16} \frac{x^2 y}{32 z^5} = \log_{16} x^2y - \log_{16} 32z^5$

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

$&= \log_{16}x^2 + \log_{16} y - \left(\log_{16} 32 + \log_{16} z^5 \right) \\&= 2 \log_{16} x + \log_{16} y - \frac{5}{4} -5 \log_{16}z$

Simplify $\log_{16} 32 \rightarrow 16^n = 32 \rightarrow 2^{4n} = 2^5$ 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).

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

$\log (5c^4)^2 &= 2 \log 5c^4 \\&= 2(\log 5 + \log c^4) \\&= 2(\log 5 + 4 \log c) \\&= 2 \log 5 + 8 \log c$

Important Note: You can write this particular log several different ways. Equivalent logs are: $\log 25 + 8 \log c, \log 25 + \log c^8$ and $\log 25c^8$ . 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.

$\ln 5 - 7 \ln x^4 + 2 \ln y &= \ln 5 - \ln x^{28} + \ln y^2 \\&= \ln \frac{5 y^2}{x^{28}}$

Important Note: If the problem was $\ln 5 - (7 \ln x^4 + 2 \ln y)$ , 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.

### Vocabulary

Power Property
As long as $b \ne 1$ , then $\log_b x^n = n \log_b x$ .

### Practice

Expand the following logarithmic expressions.

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

Condense the following logarithmic expressions.

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

### Vocabulary Language: English

Power Property

Power Property

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