Is there a closed form expression for the intersection point of a linear function and a exponential function?

Question

$$ a x = be^{cx} - 1 $$ where $a,b,c$ are constant and $a > 0, 0 < b < 1$ and $c>0$. Is there a closed form expression for this function?

Answer 1

I assume by "closed-form expression" you mean a function that takes as parameters the constants $a,b,c$ determining the linear function and the exponential and returns the $x$-value of the intersection point.

Solve[a x == b E^(c x) - 1, x]

in Mathematica yields (as pointed out in the comments)

$$ x = \dfrac{-c - a \cdot W \left( - \frac{b c\operatorname{e}^{- \frac{c}{a}}}{a}\right)}{a c},$$ where $W$ is the Lambert W-function. I hope this helps.