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

Question

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 $$ {xe^x, xe^{e^x},xe^{e^{e^x}}}... $$

Answer 1

These are so-called hyper-Lambert functions, see On some applications of the generalized hyper-Lambert functions.

Answer 2

Define $({}^1e(x),\ {}^2e(x),\ \dotsc)=(e^x,\ e^{e^x},\ \dotsc)$, and let $\phi_n$ denote the inverse of the function $x\mapsto x\ {}^ne(x)$.

Because the terms $x$, ${}^ne(x)$ are algebraically independent, we don't know how to rearrange the defining equation for $\phi_n$, $y\ {}^ne(y)=x$, algebraic over $\mathbb{C}$, where $y=\phi_n(x)$, for $x$ by only elementary functions.

The main theorem in [Ritt 1925] implies that the function term $x\ {}^ne(x)$ isn't in a form to read if a bijective elementary function $x\mapsto x\ {}^ne(x)$ has an elementary inverse or not.

The inverses were investigated, they are called Hyper Lambert W:

\begin{gather*} x\ ^ne(x)=y \\ G(\underbrace{1,1,\dotsc,1}_{\text{$(n-1)$ times}};x)=y \\ x=\operatorname{HW}(\underbrace{1,1,\dotsc,1}_{\text{$(n-1)$ times}};y). \end{gather*}

So we have a closed form for $x$, and the representations of Hyper Lambert W give hints for calculating $x$.

See also my answer to Two kind of equations involving natural log and exponentiation.

Galidakis, I. N.: On solving the p-th complex auxiliary equation $f^{(p)}(z)=z$. Complex Variables 50 (2005) (13) 977-997

Galidakis, I. N.: On some applications of the generalized hyper-Lambert functions. Complex Variables and Elliptic Equations 52 (2007) (12) 1101-1119

[Ritt 1925] Ritt, J. F.: Elementary functions and their inverses. Trans. Amer. Math. Soc. 27 (1925) (1) 68-90