Solving a transcendental equation using Lambert's W function

Question

I found that transcendental equations with this structure:

$$\ e^{-cx}=a_0(x-r_1) $$

Have this kind of Lambert-W-based solutions:

$$\ x=r_1 + \frac1c\ W \left(\frac{\ ce^{-cr_1}}{a_0}\right) $$

What is the procedure to proof that this is true? I would be thankful if you may help me figure this out.

The formulae were retrieved from: http://dx.doi.org/10.1145/2992274.2992275

Answer 1

Start with the initial equation: $$e^{-cx}=a_0(x-r_1)$$

Multiply both sides by $\frac{c}{a_0}e^{c(x-r_1)}$:

$$\frac{ce^{-cr_1}}{a_0}=(cx-cr_1)\cdot e^{cx-cr_1}$$

Apply $W$ to both sides: $$W\bigg(\frac{ce^{-cr_1}}{a_0}\bigg)=cx-cr_1$$ Solve for $x$: $$x=\frac{1}{c} W\bigg(\frac{ce^{-cr_1}}{a_0}\bigg) +r_1$$ Done!