How to solve for $x$ : $ae^{bx}+x = c$

Question

How to solve for $x$ : $ae^{bx}+x = c$

I've tried to solve it with the Lambert W function, but, in all of the methods I tried, I had $x$ as the Lambert W function parameter. Can I solve it without graphs? Thank you for your help.

Answer 1

$$ae^{bx}+x=c$$

$$ae^{bx}=c-x$$

$$ae^{bc-b(c-x)}=c-x$$

$$abe^{bc}=b(c-x)e^{b(c-x)}$$

$$b(c-x)=W(abe^{bc}).$$

$$x=c-\frac1bW(abe^{bc}).$$

(By hand.)