Possible Duplicate:
$x^y = y^x$ for integers $x$ and $y$
How to prove that $(2,4)$ and $(4,2)$ are the only solutions of Diophantine equations ${x^y} = {y^x}$ for $x \ne y$?
Possible Duplicate:
$x^y = y^x$ for integers $x$ and $y$
How to prove that $(2,4)$ and $(4,2)$ are the only solutions of Diophantine equations ${x^y} = {y^x}$ for $x \ne y$?
Taking logarithms:
$$x\ln y=y\ln x$$ $$\frac{\ln y}y=\frac{\ln x}x$$
For this to be true, the function has to take the same value at two different locations. Take the function
$$f(x)=\frac{\ln x}x$$ $$f'(x)=\frac{1-\ln x}{x^2}$$
It has a maximum at $x=e$, is decreasing for $x>e$ and increasing for $x