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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.

Answers

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

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

\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]

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Date Created:

Mar 12, 2013

Last Modified:

May 27, 2014

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