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Math Help - Solve x^y=0.5 for y.

  1. #1
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    Question [solved] Solve x^y=0.5 for y.

    Part of a project I'm working on involves calculating a gamma correction value for a color. My question is: Given x, how do I solve x^y=0.5 for y? I know that 0 < x < 1 and y > 0.

    Right now I'm doing a binary search to solve for y, but I'd rather have a direct, exact solution and I can't figure it out.

    Thanks!
    J
    Last edited by jasonc2; June 18th 2012 at 09:29 PM.
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    Re: Solve x^y=0.5 for y.

    Quote Originally Posted by jasonc2 View Post
    Part of a project I'm working on involves calculating a gamma correction value for a color. My question is: Given x, how do I solve x^y=0.5 for y? I know that 0 < x < 1 and y > 0.

    Right now I'm doing a binary search to solve for y, but I'd rather have a direct, exact solution and I can't figure it out.
    x^y = \frac12

    \Rightarrow\ln x^y = \ln\frac12

    \Rightarrow y\ln x = -\ln2

    \Rightarrow y = -\frac{\ln2}{\ln x}
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    Re: Solve x^y=0.5 for y.

    Awesome, thank you so much! =)
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    Re: Solve x^y=0.5 for y.

    Quote Originally Posted by Reckoner View Post
    x^y = \frac12

    \Rightarrow\ln x^y = \ln\frac12

    \Rightarrow y\ln x = -\ln2

    \Rightarrow y = -\frac{\ln2}{\ln x}
    Or just \displaystyle \begin{align*} y = \log_{x}{\frac{1}{2}} = -\log_x{2} \end{align*}.
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    Re: Solve x^y=0.5 for y.

    Thanks. I don't have logx available to use, only ln and log10.
    Last edited by jasonc2; June 19th 2012 at 01:33 AM.
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