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Math Help - a^x = x

  1. #1
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    a^x = x

    How can we algebraically find the intersection of an exponential function y=a^x with the straight line y=x? In other words, what is the solution to the equation a^x = x ?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by hakanaras View Post
    How can we algebraically find the intersection of an exponential function y=a^x with the straight line y=x? In other words, what is the solution to the equation a^x = x ?
    First there is no guarantee that there are any real solutions to a^x=x. In fact if \ln(a)>1 it is obvious that there are no real solutions.

    If a<1, then put b=1/a and the equation becomes:

    e^{-\ln(b)x}=x

    which we can rearrange to:

    [\ln(b)~x]~e^{\ln(b)~x}=\ln(b)

    and this has solution:

    x=\frac{W(\ln(b))}{\ln(b)}

    or equivalently:

    x=-~\frac{W(-\ln(a))}{\ln(a)}

    where W is Lambert's W function.

    (If we are not concerned with real solutions then the above works period, though you have to keep an eye on the multiple values of the W function)

    RonL
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  3. #3
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    Quote Originally Posted by CaptainBlack View Post

    :
    :

    or equivalently:

    x=-~\frac{W(-\ln(a))}{\ln(a)}

    where W is Lambert's W function.

    (If we are not concerned with real solutions then the above works period, though you have to keep an eye on the multiple values of the W function)

    RonL
    To demonstrate this working see this:

    Code:
    This is EULER, Version 2.3 RL-06.
    
    Type help(Return) for help.
    Enter command: (16777216 Bytes free.)
    
    Processing configuration file.
    
    Done.
    >load "C:\Program Files\EulerRL\Euler Utils\LambertW.e";
    >
    >a=0.1;
    >x=-LambertW(-log(a))/log(a)
         0.399013 
    >
    >
    >
    >a^x-x
     5.55112e-017 
    >
    RonL
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  4. #4
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    Some replies do not deserve an unthanked fate. For my money, the above two are in that class. Hence the pre-emptive safe-guard.

    (Would've added rep but apparently I've got to spread a bit more first).

    And this thread is certainly as good as any to direct the interested reader to the following references:

    http://www.cs.uwaterloo.ca/research/tr/1993/03/W.pdf

    New analytic solution to classic problem?
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  5. #5
    Grand Panjandrum
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    Quote Originally Posted by mr fantastic View Post
    Some replies do not deserve an unthanked fate. For my money, the above two are in that class. Hence the pre-emptive safe-guard.

    (Would've added rep but apparently I've got to spread a bit more first).

    And this thread is certainly as good as any to direct the interested reader to the following references:

    http://www.cs.uwaterloo.ca/research/tr/1993/03/W.pdf

    New analytic solution to classic problem?
    Now you have embarrassed me

    Great references.

    RonL
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