Solving the transcendental equation $\operatorname{Li}_{3}(e^{-kx}) + x \operatorname{Li}_{2}(e^{-kx}) = k x^3$.

I need to solve the following equation:

$$\operatorname{Li}_{3}(e^{-kx}) + x \operatorname{Li}_{2}(e^{-kx}) = k x^3$$

for $x\in\mathbb{R}^{\ast}$ and where $k\in\mathbb{R}^{+}$. Here $\operatorname{Li}_{3}$ and $\operatorname{Li}_{2}$ are the third and second polylogarithms. I have been able to show that the solution exists and it is unique, but I do not know how to continue further. In fact I am wondering about the very meaning of "solving the equation". What means solving an equation like this? I don't expect to solve it in terms of known transcendental functions, which takes me to my other question:

Is the inverse polylogarithm $\operatorname{Li}^{-1}_{n}(z)$ defined? It is possible to solve the equation in terms of inverse polylogarithms?

Thanks.