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Just wondering about x^y = y^x

  1. Nov 22, 2006 #1
    This wasnt set for homework just something i thought up so i hope this is the right place to post?

    Anyway when i put x^y = y^x (or also lnx/lny = x/y) in a graphing program i realised that there were two lines, y = x and another line that looked a bit like y = e^(e/x) but wasnt.

    So this kind of interested me... and i have two questions:

    can y^x = x^y be written as y = f(t) and x = g(t) and if so what are f(t) and g(t)?

    Is it possible to find the equation of just the line that looks like y = e^(e/x) ?

    I have thought about it a bit but have no idea how to go about solving either question. Any degree level maths will be beyond me (doing A level atm) so hopefully there is something simple that i have overlooked?
  2. jcsd
  3. Nov 22, 2006 #2


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    How did you put xy= yx into a graphing program? Most graphing programs require that the function be put in the form y= f(x) which is the whole problem here.

    Solving yx= xy, or, equivalently, y= xy/x for y looks to me like it would involve the "Lambert W function", defined as the inverse to the function f(x)= xex.
  4. Nov 22, 2006 #3


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    There was an old thread about this. One thing you can do is graph in 3-d z = x^y - y^x
  5. Nov 22, 2006 #4
    I used a trial version of derive 6 and plugged x^y=y^x straight in. It only plotted for positive values and went a bit crazy at high values of x and y. I think it was fairly accurate though. I remember putting one of the values for the non x=y line into my calculator and it working.
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