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Math Help - Solving for x in logarithm problem

  1. #1
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    Post Solving for x in logarithm problem

    The equation (attached as image) has
    (a)one irrational solution
    (b)no prime solution
    (c)two real solutions
    (d)one integral solution

    i would like to get help on how to find the possible values of x

    i solved the equation by applying the base changing property and then doing
    L.C.M of both equations but after that i am unable to solve further?

    Any help will be highly appreciated.
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  2. #2
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    Re: Solving for x in logarithm problem

    Show what you tried so far.

    What results?

    I suggest: Do change of base with base of 2 .
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  3. #3
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    Re: Solving for x in logarithm problem

    2/log2(x) + 6/ (log2(x)+1)=3
    i got this result
    after that what should i do
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  4. #4
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    Re: Solving for x in logarithm problem

    Solve for log_2 (x) .
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  5. #5
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    Re: Solving for x in logarithm problem

    Quote Originally Posted by sumedh View Post
    The equation (attached as image) has
    (a)one irrational solution
    (b)no prime solution
    (c)two real solutions
    (d)one integral solution

    i would like to get help on how to find the possible values of x

    i solved the equation by applying the base changing property and then doing
    L.C.M of both equations but after that i am unable to solve further?

    Any help will be highly appreciated.
    I assume that the question reads:

    \log_{x^2}(16)+\log_{2x}(64)=3

    If so:

    2. Use the base change formula:

    \dfrac{4\ln(2)}{\ln(x^2)}+\dfrac{6\ln(2)}{\ln(2x)}  =3

    \dfrac{4\ln(2)}{2\ln(x)}+\dfrac{6\ln(2)}{(\ln(2)+\  ln(x))}=3

    3. Multiply through by ln(x):

    2\ln(2)+\dfrac{6\ln(2) \ln(x)}{(\ln(2)+\ln(x))}=3 \ln(x)

    Multiply through by (ln(2)+ln(x)):

    2 \ln(2) (\ln(2)+\ln(x))+6\ln(2) \ln(x)=3 \ln(x)(\ln(2)+\ln(x))

    4. Expand the brackests and collect like terms. You'll get a quadratic equation in ln(x). Solve for ln(x) and consequently for x.

    5. For confirmation only: There are 2 solutions: x = 4 ~\vee~x=\frac12 \cdot \sqrt[3]{4}
    Last edited by earboth; August 12th 2011 at 11:27 AM.
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  6. #6
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    Re: Solving for x in logarithm problem

    thank you very much i got it
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