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Math Help - Power Law for Logarithms

  1. #1
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    Power Law for Logarithms

    Solve for x in each of the following:

    log_2x = 3log_24

    My answer: 64

    --

    logx = \frac{1}{3}log8

    My answer: 2


    Are my answers correct?



    Evaluate each of the following:

    1) log_2 \sqrt [5] {16}

    2) log_525 \sqrt [3] {5}

    Can you please show me how to solve these logs, I don't know where to start
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Macleef View Post
    Solve for x in each of the following:

    log_2x = 3log_24

    My answer: 64

    --

    logx = \frac{1}{3}log8

    My answer: 2


    Are my answers correct?
    Yes.

    Quote Originally Posted by Macleef View Post
    Evaluate each of the following:

    1) log_2 \sqrt [5] {16}

    2) log_525 \sqrt [3] {5}

    Can you please show me how to solve these logs, I don't know where to start
    We know that
    log_b(a^x) = x \cdot log_b(a)

    So for the first one:
    log_2 ( \sqrt[5]{16} ) = \frac{1}{5} \cdot log_2(16) = \frac{1}{5} \cdot 4 = \frac{4}{5}

    You do the second one.

    -Dan
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  3. #3
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    For the second equation, is the following what you do...

    = \frac {1}{3} (log_525(5))

    = \frac {1}{3} (log_5 5^{2} (5^{1}))

    = \frac {1}{3} (2) (1)

    = \frac {2}{3}
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  4. #4
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Macleef View Post
    For the second equation, is the following what you do...

    = \frac {1}{3} (log_525(5))

    = \frac {1}{3} (log_5 5^{2} (5^{1}))

    = \frac {1}{3} (2) (1)

    = \frac {2}{3}
    No, here we have another rule coming into play as well:
    log_b(xy) = log_b(x) + log_b(y)

    So....
    log_5(25\sqrt[3]{5})

    = log_5(25) + log_5(\sqrt[3]{5})

    Can you finish from here?

    -Dan
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  5. #5
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    Quote Originally Posted by topsquark View Post
    No, here we have another rule coming into play as well:
    log_b(xy) = log_b(x) + log_b(y)

    So....
    log_5(25\sqrt[3]{5})

    = log_5(25) + log_5(\sqrt[3]{5})

    Can you finish from here?

    -Dan

    Is the answer \frac {7}{3}?


    My work:

    = log_55^{2} + \frac {1}{3} log_55

    = 2(log_55) + \frac {1}{3} log_55

    = 2 + \frac {1}{3}

    = \frac {7}{3}
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  6. #6
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Macleef View Post
    Is the answer \frac {7}{3}?


    My work:

    = log_55^{2} + \frac {1}{3} log_55

    = 2(log_55) + \frac {1}{3} log_55

    = 2 + \frac {1}{3}

    = \frac {7}{3}
    Yup!

    -Dan
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