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Math Help - Logarithm Problem I can't figure out

  1. #1
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    Logarithm Problem I can't figure out

    These questions are from several math contests, and I'm trying to do these but I can't figure it out.

    1. If x and y > 0 , log(base y)x + log(base x)y = 10/3 ,and (x)(y)=144
    ,find x + y /2

    2. How many real numbers x satisfy the equation (1/5) log(base 2)x = sin(5pi x) ?

    I tried making y=144/x and plugging that into the equation, but I have no idea what to do for the second question

    Any help from you guys is greatly appreciated
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  2. #2
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    Quote Originally Posted by ConMan View Post
    These questions are from several math contests, and I'm trying to do these but I can't figure it out.

    1. If x and y > 0 , log(base y)x + log(base x)y = 10/3 ,and (x)(y)=144
    ,find x + y /2

    ...
    1. You are supposed to know that \log_b(a)=\dfrac1{\log_a(b)}

    2. Using this rule your equation becomes:

    \log_y(x)+\dfrac1{\log_y(x)} = \dfrac{10}3

    Use the substitution z = \log_y(x) . Then you have:

    z-\dfrac1z = \dfrac{10}3~\implies~z^2-\dfrac{10}3 z + 1 = 0

    which yields z = 3~\vee~z = \dfrac13

    3. Re-substituting yields:

    x = y^3~\vee~x = \sqrt[3]{y}

    4. Now use the second equation x\cdot y = 144 to calculate the values of x and y.
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  3. #3
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    Thank you for your timely reply earboth, I really appreciate your help

    Also, does anyone know about some online resources regarding some properties/rules on logarithms?

    EDIT: this is nice: http://en.wikipedia.org/wiki/List_of...mic_identities
    Last edited by ConMan; November 4th 2009 at 02:48 PM.
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  4. #4
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    Logarithm - Wikipedia, the free encyclopedia

    Wikipedia, man's best friend!
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  5. #5
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    Okay scratch the second question, since sin of (pi) is equal to zero. (Doh!)
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