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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?