I would like to know how can I solve the following exponential equation for $x$: $$\exp\left(\frac{n_1}{x}\right) + \exp(n_2) + \exp\left(n_3 - \frac{n_4}{x}\right) - \exp(n_5) = 0$$ where $n_1$, $n_2$, $n_3$, $n_4$, and $n_5$ are constants.
Many thanks in advance!
Let $y=\exp(1/x)$ and $n_i'=\exp(n_i)$ and multiply both sides by $y^{n_4}$ to get
$$y^{n_1+n_4}+(n_2'-n_5')y^{n_4}+n_3'=0$$
which is a trinomial in $y$, which cannot be algebraically solved in general. It can, of course, be solved numerically.