I am looking for the solution of an equation just like
$$ae^{k_{1}x} - be^{k_{2}x} + c = 0$$
where $a$, $b$, $k_{1}$, $k_{2}$ are known real constants, $x$ is unknown. I already have a numerical solution but for optimization purpose and generalization I would need to have a closed formula. It would be nice to have a solution based on well known and already well implemented functions such as Lambert function, or generalized hyper-geometric . Any opinion or clue on the way to solve such equation would be very useful.
\begin{align} a\exp(k_1x) - b\exp(k_2x) + c &= 0 \tag{1}\label{1} \end{align}
The expression \eqref{1} is equivalent to
\begin{align} p\,z^u-q\,z-1&=0 \tag{2}\label{2} \\ \text{for }\quad z&=\exp(k_1\,x) ,\\ u&=\frac{k_2}{k_1} ,\quad p=\frac bc ,\quad q=\frac ac . \end{align}
Unfortunately, it is known that equation of the form \eqref{2} does not have a closed-form solution, not even in terms of the Lambert W function, unless $u$ happens to be a small integer less than $5$ hence, the numerical root finding is the only option in this case.