How should determine solutions to equations of this form? 
$$e^{-f(x)} + b f(x) = ax$$
Here $f(x)>0$ is real valued. Also $a>0$, $b>0$.



How to solve $e^{f(x)} + a f(x) + bx = 0$
$f(x) = W(-\exp(-ax/b)/b) + ax/b$
where W is one of the branches of the [Lambert W function](http://en.wikipedia.org/wiki/Lambert_W_function).


