Inverse function of $y=\frac{\ln(x+1)}{\ln x}$

Well, as you know, I can't actually give you a nice solution to put a box around, but I can give you some information about the function $f(x)$ satisfying $f(x)^x+f(x)+1=0$.

There is a pleasant, if not closed-form, asymptotic expansion for $f(x)$ for large $x$ in polynomials of $s=\log 2$:

$$\begin{align} f(x)=1+\frac sx+&\frac1{2x^2}\left(s+s^2\right)+\frac1{24x^3}\left(6 s+15 s^2+4 s^3\right)+ \\ +&\frac1{48x^4} \left(6 s+27 s^2+18 s^3+2 s^4\right)+\cdots \end{align}$$

This was obtained by series inversion on the series for $\dfrac{\log(x+1)}{\log x}$ about $x=1$. More specifically, letting $y=x-1$, the series for

$$\begin{align} \frac1{f^{-1}(x)} & =\frac{\log x}{\log(x+1)}=\frac{\log(y+1)}{\log(y+2)}=\frac{\log(y+1)}{\log(y/2+1)+s} \\ & =\frac{-\sum_1^\infty(-y)^n/n}{s-\sum_1^\infty(-y/2)^n/n}:=\sum_1^\infty a_ny^n \end{align}$$

can be solved for $a_n$ using the Cauchy product formula $\sum_0^\infty \alpha_ny^n\sum_0^\infty \beta_ny^n=\sum_{n=0}^\infty[\sum_{k=0}^n\alpha_k\beta_{n-k}]y^n$ to get

$$a_n=\frac{(-1)^{n+1}}{sn}+\sum_{k=1}^{n-1}\frac{(-2)^{-k}a_{n-k}}{sk}$$

(which yields coefficients in $\mathbb Q[1/s]$) and if the inverse of this is $f(x)=\sum_0^\infty b_nx^{-n}$, then

$$\begin{align} y+1 & =f(f^{-1}(y+1))=\sum_{n=0}^\infty b_n\bigg(\sum_{k=1}^\infty a_ky^k\bigg)^n=\sum_{n=0}^\infty b_n\bigg(\sum_{k=1}^\infty a_k(x+1)^k\bigg)^n \\ & =b_0+(a_1b_1)y+(a_2b_1+a_1^2b_2)y^2+(a_3b_1+2a_1a_2b_2+a_1^3b_3)y^3+\cdots. \end{align}$$

Unfortunately, the last sum has no nice closed form (although it can be done, using Faà di Bruno's formula), and the solution can only be written recursively anyway, so I'll stop here. This equation is supposed to be solved component-by-component to get the terms in the formula, so $1=b_0$, $1=a_1b_1\Rightarrow b_1=s$, $0=a_2b_1+a_1^2b_2\Rightarrow b_2=(s+s^2)/2$, etc. This is the source of the series quoted above.

The series is convergent for $|x|>1$. Here is a plot of the function, in the complex plane:

Red means near $1$, yellow-green is near $i$, blue is near $-i$, white is near $\infty$, black is near $0$. And here is a plot of the above asymptotic expansion, up to the $x^{-12}$ term:

A basic approximation of the function around $x=0$ is given by

$$x=\frac{\log(y+1)}{\log y}=\frac y{\log y}+O(xy)\Rightarrow -\frac1x=-\frac yxe^{-y/x}=y\Rightarrow y=-xW\Big(\!-\frac1x\!\Big),$$

where $W(x)$ is the Lambert W-function. The asymptotics of this function near $\infty$ are quite complex, and there is no proper power series, or even a generalized asymptotic series (involving terms other than $x^k$ for $k\in\mathbb Z$), because the truncated series, with error term $O\left(\big(\frac{\log\log x}{\log x}\big)^m(\log x)^{-k}\right)$, is never $O(1/x)$. The "basic" solution $y=-xW(-1/x)$ is however accurate to $O(x^2)$ (or so, proof or revision to come).