I am a physical chemist, and we are trying to model a simple process. We ended up arriving at the following equation.
\begin{align} ax^{-7/2}e^{-bx(ln(cx)-1)}=1 \end{align}
We'd like to solve for $x$. All constants ($a$, $b$, $c$) are real, positive. a is on the order of 31 and b and c are on the order of the -3. I have found the following relationship from the following: (Edwards, https://arxiv.org/pdf/1902.08910.pdf)
\begin{align} y=ax^be^{cx^d}+f \end{align}
Can be inverted into
\begin{align} x=\left[\frac{b}{cd} W\left(\frac{cd(y-f)^{\frac{d}{b}}}{ba^{\frac{d}{b}}}\right)\right]^{1/d} \end{align} Where W is the Lambert W function.
But obviously I have a $\ln(x)$ in the exponential function. We'd love to find a good expression for x, either exact, or approximate through expansion. We can check it against a numerical solution.
Any ideas? Thanks all!