Just wondering about x^y = y^x

[Sink41]

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 realized 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?

 

[HallsofIvy]

How did you put x^y= y^x 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 y^x= x^y, or, equivalently, y= x^y/x for y looks to me like it would involve the "Lambert W function", defined as the inverse to the function f(x)= xe^x.

 

[0rthodontist]

There was an old thread about this. One thing you can do is graph in 3-d z = x^y - y^x

 

[Sink41]

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.