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Math Help - Solve an equation system with logarithms

  1. #1
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    Solve an equation system with logarithms

    x^log(y)+y^log(x^(1/2))=110
    xy=1000

    This is my attempt:
    log(y)*log (x)+log (x^(1/2)*log(y)=log 110
    y=1000/x

    log(1000/x)*log (x)+log (x^(1/2)*log(1000/x)=log 110
    log (1000)-log (x)*log (x)+log (x^(1/2)*log(1000)-log(x)=log 110

    log(x)=t
    log (1000)=3

    (3-t)t+(t/2)(3-t)=log 110
    3t^-3t^2+(3t-t^2)/2=log 110

    I believe this is an incorrect solution, can you please tell where my error is? If my attempt is incorrect from the beginning, please show me how to solve it. Thank you in advance!
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  2. #2
    Member anonimnystefy's Avatar
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    Re: Solve an equation system with logarithms

    hi anna

    x^{\log(y)}+y^{\log(x^{\frac{1}{2}})}=110

    y^{\log(x)}+y^{\log(x^{\frac{1}{2}})}=110

    (y^{\log(x^{\frac{1}{2}})})^2+y^{\log(x^{\frac{1}{  2}})}=110

    \text{Let: } k=y^{\log(x^{\frac{1}{2}})}

    \text{You get: }k^2+k=110

    can you continue it now?
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  3. #3
    Junior Member
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    Re: Solve an equation system with logarithms

    Thank you for your solution!

    k=-11, 10

    10=y^log (x^(1/2))
    log 10=(1/2) log x*log y
    2=lg x*2 log y
    2=log x*log y^2

    Is this correct? If not, please show me the correct solution. Thank you!
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