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Math Help - Log question- need help

  1. #1
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    Log question- need help

    Hey guys.. I have a log question that I am working on for something I need to hand in today..

    I have the answer, I was helped by someone, I am just having trouble understanding a step in the answer, and I don't really want to just move on without being able to grasp whats happening.

    Here is the question:

    Image of Logs question - Photobucket - Video and Image Hosting

    I felt this would be faster then to write it out..
    I am wondering what is happening in the 3rd to 4th step mainly.. how does the -1 exponent become -3/2, and what happened to the square root sign and how did that interact?

    Thanks alot
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  2. #2
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    Note that square roots can be written using fractional exponents: \sqrt{a} = a^{\frac{1}{2}}

    More generally: \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m = a^{\frac{m}{n}} (if there is no n written, it is implied to be 2)

    Here, imagine: \color{red} a = \tfrac{2}{3}, \color{magenta} n = 2 and \color{blue} m = 3

    So:

    \begin{aligned} 2 + \log_{\frac{2}{3}} \left( {\color{magenta}\sqrt{{\color{black}\bigg( }{\color{red}\frac{2}{3}}{\color{black}\bigg)^{{\c  olor{blue}3}}}}}\right)^{-1} & = 2 + \log_{\frac{2}{3}} \left[\left(\frac{2}{3}\right)^{\frac{3}{2}}\right]^{-1} \\ & = 2 + \log_{\frac{2}{3}}\left(\frac{2}{3}\right)^{-1} \qquad \text{Since: } \left(a^b\right)^c = a^{bc} \\ & \ \ \vdots \end{aligned}
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  3. #3
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    Hello, Slipery!

    \log_{\frac{2}{3}}\!\left(\frac{4}{9}\cdot\sqrt{\f  rac{27}{8}}\right)
    I would do it like this . . .

    \log_{\frac{2}{3}}\! \left[\frac{2^2}{3^2}\cdot\left(\frac{3^3}{2^3}\right)^{  \frac{1}{2}} \right] \;\;=\;\;\log_{\frac{2}{3}}\!\left(\frac{2^2}{3^2}  \cdot\frac{3^{\frac{3}{2}}}{2^{\frac{3}{2}}}\right  ) . = \;\;\log_{\frac{2}{3}}\!\left(\frac{2^{\frac{1}{2}  }}{3^{\frac{1}{2}}}\right) \;\;=\;\;\log_{\frac{2}{3}}\!\left(\frac{2}{3}\rig  ht)^{\frac{1}{2}}


    . . . =\;\;\frac{1}{2}\cdot\underbrace{\log_{\frac{2}{3}  }\!\left(\frac{2}{3}\right)}_{\text{This is 1}} \;\;=\;\;\frac{1}{2}\cdot1 \;\;=\;\;\frac{1}{2}

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  4. #4
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    You were both very helpful! kind of nice seeing two ways it can be done.

    Thank you so much
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