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Thread: type of exponential equation

  1. #1
    Senior Member tukeywilliams's Avatar
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    type of exponential equation

    To solve $\displaystyle 2^x = x^5 $ could you use Lambert W function?

    Thanks
    Last edited by tukeywilliams; Jul 29th 2007 at 08:57 PM.
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  2. #2
    Senior Member tukeywilliams's Avatar
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    It can be solved via derivatives and intersections of curves. But how would you solve it in a purely algebraic manner using the W Lambert function?
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by tukeywilliams View Post
    It can be solved via derivatives and intersections of curves. But how would you solve it in a purely algebraic manner using the W Lambert function?

    In principle yes it can, the equation can be transformormrd to:

    $\displaystyle
    \frac{-x \ln(2)}{5}e^{\frac{-x \ln(2)}{5}}=\frac{- \ln(2)}{5}
    $

    So formaly we have:

    $\displaystyle
    \frac{-x \ln(2)}{5} = W\left(\frac{- \ln(2)}{5}\right) \approx W(-0.1386)
    $

    Now the problem is that there a multiple (real) branches of the Lambert W in
    the region around $\displaystyle -0.1386$, and you are going to have to
    evaluate two of them to get all the solution to this. My LambertW calculator
    gives $\displaystyle x\approx 1.17724$.

    However just plot the functions and you will see that there is another
    solution near $\displaystyle x=22$ Numerical methods will find both solutions
    once you know where they are.

    RonL
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  4. #4
    Bar0n janvdl's Avatar
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    Here is a graph if you'd like to see...
    Attached Thumbnails Attached Thumbnails type of exponential equation-graph2.jpg  
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  5. #5
    Bar0n janvdl's Avatar
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    Quote Originally Posted by CaptainBlack View Post

    ...there is another solution near $\displaystyle x=22$...

    RonL
    I really couldn't see that one when looking at the graph...
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  6. #6
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by janvdl View Post
    I really couldn't see that one when looking at the graph...
    A binary search gives me about x = 22.440006

    -Dan
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