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Math Help - Logarithmic Equation

  1. #1
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    Logarithmic Equation

    I have a problem, whose solution, apperantly, eludes me.
    I have chosen to upload it as a .gif file. The second equation is what I get after the obvious substitution and simplification. This is not correct, the answer is suppsed to be t=3, hence x = 1000, and I fail to find my mistake, which is why I am asking for help.
    Thank you.
    Attached Thumbnails Attached Thumbnails Logarithmic Equation-lgx.gif  
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  2. #2
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    \sqrt{1 + \log{x}} + \sqrt{2\log{x} - 2} = 4

    let t = \log{x}

    \sqrt{1 + t} + \sqrt{2t - 2} = 4

    (1+t) + 2\sqrt{(1+t)(2t-2)} + (2t-2) = 16

    2\sqrt{(1+t)(2t-2)} = 17-3t

    4(1+t)(2t-2) = (17-3t)^2

    8t^2 - 8 = 9t^2 - 102t + 289

    0 = t^2 - 102t + 297

    0 = (t - 3)(t - 99)

    t = 3, t = 99

    log(x) = 3, x = 1000

    log(x) = 99 is an extraneous solution
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  3. #3
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    Thank you. I have now understood my mistake. A silly one it was, as expected.

    Your time is greatly appreciated.
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