If $x^y = xy$, what is $y$ in terms of $x$?

Question

I know that $x=y^{\frac{1}{y-1}}$ but I cannot solve for $x$. Can somebody please help me? It should be possible because it is a very simple equation, can somebody please solve it.

Answer 1

As I say in the comments, many things are undefined, but if $x$ never disappears, then we can write $$y=x^{y-1},$$ which is equivalent to $$y-1=\log_x y.$$ We then see that $x,y$ should satisfy the conditions $1\ne x>0,y>0$ for uniquely defined values of $y$ regarded as a real-valued function of the real variable $x.$

Also, it appears that there's no elementary way to express $y$ in terms of $x.$