How can I solve $x = e^{a+bx} + c?$

Question

I need to solve this implicit equation for a physical system. I know that the similar equation $x = xe^x$ (solved with the Lambert W-function) doesn't have real roots, but since this equation has additional constants it should have real roots. However, my background in these type of equations isn't great so I would appreciate your help.

Answer 1

Rewrite it as

$$b(x-c) = be^{a+bc} e^{b(x-c)}$$

$$-b(x-c) e^{-b(x-c)} = -be^{a+bc}$$

$$-b(x-c) = W(-be^{a+bc})$$

$$x = c - \frac{1}{b} W(-be^{a+bc})$$