Solving Lambert style equation

Question

I am looking for the values $ x \in R $ which satisfy the following equation :

$ e^{-\alpha x} = \frac{a}{x - c} $

Where $ \alpha $, $ a $ and $ c $ are real valued constants.

If $ c = 0 $, we get $ x = - \frac{W(-a \alpha)}{\alpha} $, where $ W$ denotes the Lambert W function, but with $ c \neq 0 $ I don't see an obvious solution.

Otherwise, could I find an approximate solution with numerical methods in limited time ? (my system needs to be running in real time)

Answer 1

Doing what @user721481 suggested, you should arrive at $$x=c-\frac{W\left(-a \alpha e^{\alpha c}\right)}{\alpha }$$

Answer 2

Step by step: $$ e^{-\alpha x} = \frac{a}{x-c} \\ (x-c)e^{-\alpha x} = a \\ (x-c)e^{-\alpha x} e^{\alpha c} = a e^{\alpha c} \\ (x-c)e^{-\alpha (x-c)} = a e^{\alpha c} \\ -\alpha(x-c)e^{-\alpha (x-c)} = -a\alpha e^{\alpha c} \\ -\alpha(x-c) = W(-a\alpha e^{\alpha c}) \\ x-c = -\frac{W(-a\alpha e^{\alpha c})}{\alpha} \\ x = c-\frac{W(-a\alpha e^{\alpha c})}{\alpha} $$