Solve $\ln(1-x)-\ln(x)+\frac{a}{x}=c$ for $x$

Question

Is there an analytical (maybe involving special functions) solution of an equation of the form:

$$\ln(1-x)-\ln(x)+\frac{a}{x}=c$$

Here I want to solve for $x$, which should satisfy $0\le x\le1$, and $a,c$ are real constants.

Answer 1

The solution can be written using the Lambert W function as

$$ x = \frac{a}{W(a e^{c-a})+a} $$

If $-e^{-1} < a e^{c-a} < 0$ there will be two real solutions, one using the principal branch and one using the "$-1$" branch of W.