How can I find the roots of this for k?

Question

$$ln(1-e^{-kx})(1-e^{-kx})+kxe^{-kx}=0$$

I need to find $k$ in this equation , it should be a function of $x$. Any hints on how should I do it ?

Answer 1

If $t = e^{-kx}$, your equation says $$ (1-t) \ln(1-t) = t \ln(t) $$ Rather obviously $t=1/2$ is a solution. There are no other real solutions (unless you count $t=0$ and $t=1$).