Inverse of $x\log(x)$ for $x>1$
The Lambert $W$ function is the inverse function of $g(x)=xe^x$, i.e. a function such that $W(x)\,e^{W(x)}=x$ for every $x$ in some range. To solve: $$ y \log y = x $$ by setting $y=e^{f(x)}$ is the same as solving $f(x) e^{f(x)}=x$, that gives $f(x)=W(x)$. It follows that: $$ y = e^{W(x)} = \frac{x}{W(x)}.$$