# Inverse of $x\log(x)$ for $x>1$

Let $y(x)=x\log(x)$ for $x>1$. Is it possible to write down the inverse function explicitly? Has this inverse function been named? (For example, the Bessel functions are "named" but cannot be defined explicitly using elementary functions.)

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: $$y = e^{W(x)} = \frac{x}{W(x)}.$$
• Okay, this makes sense to me now - the crux is the substitution $y=e^{f(x)}$. Thanks. – FreshAir Oct 13 '15 at 1:52