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Math Help - type of exponential equation

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

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

    Thanks
    Last edited by tukeywilliams; July 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:

    <br />
\frac{-x \ln(2)}{5}e^{\frac{-x \ln(2)}{5}}=\frac{- \ln(2)}{5}<br />

    So formaly we have:

    <br />
\frac{-x \ln(2)}{5} = W\left(\frac{- \ln(2)}{5}\right) \approx W(-0.1386)<br />

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

    However just plot the functions and you will see that there is another
    solution near 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 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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