# Expressions for the inverse function of f(x) = ln(x)e^x

Can the inverse of $$ln(x)e^x$$ be finitely expressed in terms of the Lambert-W function or any other well-known transcendental functions? It is clear that a closed-form elementary function expression is unreachable.

The reason I ask is in pondering on the links between the inverse Lambert-W and some naturally arising functions of similar forms. Recall that the Lambert-W, a transcendental function, is defined as $$W(xe^x) = x.$$

It is natural then to consider the inverse of functions such as $$g(x) = xe^{e^x}$$ and those with further exponentiation. With a simple transformation $$z= e^x$$ we can reduce $$g(x)$$ to the form $$ln(z)e^z$$ as originally posed. So the broader question arises: are there tangible algebraic links between the inverses of the set $${xe^x, xe^{e^x},xe^{e^{e^x}}}...$$