Prove that if x^y = y^x, then ln(x)/x = ln(y)/y.

It is given that x^y =
y^x

Take the logarithm to the base e, or ln, of both the
sides

=> ln [ x^y] = ln [
y^x]

Use the property of log that log a^b = b*log
a

=> y*ln x = x*ln
y

divide both the sides by
xy

=> [y*ln x]/xy = [x*ln
y]/xy

=> (ln x)/x = (ln
y)/y

If x^y = y^x, then
this proves that (ln x)/x = (ln
y)/y