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Math Help - x^2=2^x

  1. #1
    Super Member bigwave's Avatar
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    x^2=2^x

    x^2=2^x

    from observation 2 answers of this are 2 and 4 but x=-0.0767 is the one which not sure how got.

    if I take the log of each side 2log(x)=xlog(2) but don't see what you can do with this? thnx ahead for help
    Last edited by bigwave; November 23rd 2011 at 12:33 PM. Reason: latex
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  2. #2
    MHF Contributor Siron's Avatar
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    Re: x^2=2^x

    It's very hard to solve, 2 is indeed a solution by observation but I think the best way to determine the other solutions is by using the graph of the 2 functions.
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  3. #3
    MHF Contributor alexmahone's Avatar
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    Re: x^2=2^x

    Quote Originally Posted by bigwave View Post
    x^2=2^x

    from observation 2 answers of this are 2 and 4 but x=-0.0767 is the one which not sure how got.

    if I take the log of each side 2log(x)=xlog(2) but don't see what you can do with this? thnx ahead for help
    You need to use Newton's method or a computer program (like MATLAB).
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  4. #4
    Super Member bigwave's Avatar
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    Re: x^2=2^x

    I did use Wolfram|Alpha but they used the product log function? which I have never used. apparently one can only graph and estimate. or close in like Newton.
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  5. #5
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    Re: x^2=2^x

    I remember trying to prove as a first-year student that the equation a^x=x^a, where a is a positive real number, has exactly two positive real solutions whenever a\neq e. I don't remember solving it so I must have failed.
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  6. #6
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    Re: x^2=2^x

    Quote Originally Posted by bigwave View Post
    x^2=2^x

    from observation 2 answers of this are 2 and 4 but x=-0.0767 is the one which not sure how got.

    if I take the log of each side 2log(x)=xlog(2) but don't see what you can do with this? thnx ahead for help
    In general, this kind of equation is solved thanks to numerical computation.
    In case of x=2^x the roots are :
    x = 2 (obvious)
    x = -0.766664695962123..
    In the general case, the roots cannot be expressed in form of a combination of a finit number of elementary functions. Analytically, the solutions are expressed thanks to a special function, the Lambert function W(X)
    x = -2 W(X)/ln(2) where X= -ln(2)/2
    Since the Lambert function is multi-valuated two values are obtained for x.

    More generally the roots of equation a(x^b)=c^x, with a,b,c constant parameters, are :
    x = -b W(X)/ln(c) where X = - ln(c) / (b (a^(1/b)))
    Depending on the values of the parameters, there are two, or one, or no solution(s).
    Last edited by JJacquelin; November 25th 2011 at 07:13 AM.
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  7. #7
    Super Member bigwave's Avatar
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    Re: x^2=2^x

    well for whatever it is worth here is a graph of  x^2=2^x
    Attached Thumbnails Attached Thumbnails x^2=2^x-x-2-2-x.gif  
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  8. #8
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    Re: x^2=2^x

    Deciding where to start looking can be a bit of a trick. In this case, reasonable results can be obtained by a quick linearization or maybe a quadratic, since x^2 is quadratic already.

    x^{2} = 1 + x\cdot log(2) Produces x = 1.405 and x = -0.712

    If we move up to the quadratic...

    x^{2} = 1 + x\cdot log(2) + \frac{1}{2}\cdot\left(x\cdot log(2)\right)^{2}

    We get x = 1.691 and x = -0.778

    Moving up to the cubic is a WHOLE LOT MORE WORK, but does manage an approximation of all three solutions,

    x = 1.871, x = -0.765, and x = 12.582

    That third one isn't pretty, but it could lead to something.
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