Solve $x\log x=y$

Question

I have the following equation, $x\log x=y$.

Is it possible to solve $x$ in terms of $y$.

I think it is not possible but I am not sure.

Answer 1

$x$ must be positive, for the $\log$ to exist, so you can write $x=e^u$ and the equation becomes:

$$ue^u=y$$

This equation is well known, and has no solution in elementary function, but in terms of the Lambert W function:

$$u=\mathrm{W}(y)$$

Hence

$$x=e^{\mathrm{W}(y)}$$