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Math Help - Logarithmic equations

  1. #1
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    Logarithmic equations

    Two similar logarithmic equations:
    1) x^{lgx}=1000x^2
    2) x^{\frac{lgx+7}{4}}=10^{lgx+1}

    I tried solving the first one:
    x>0;
    I tried to write the first exponent in the base x, so that I could apply the rule a^{loga(b)}=b:
    x^{\frac{logx(x)}{logx(10)}}=1000x^2
    But it didn't work, because I got in the exponent \frac{1}{logx(10)}
    How do I solve these?
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  2. #2
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    Quote Originally Posted by Evaldas View Post
    Two similar logarithmic equations:
    1) x^{lgx}=1000x^2
    Take lgx to be \log_{10}(x)
    Then the equation becomes \left(\log(x)\right)^2-2\log(x)-3=0.
    Let u=\log(x).
    Can you solve u^2-2u-3=0~?
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  3. #3
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    Quote Originally Posted by Plato View Post
    Then the equation becomes \left(\log(x)\right)^2-2\log(x)-3=0.
    How do you get this?
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  4. #4
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    Quote Originally Posted by Evaldas View Post
    How do you get this?
    \log\left(x^{\log(x)}\right)=\log\left(1000x^2\rig  ht).
    Apply the rules of logarithms.
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