I do the calculus exercise.  You wonder about the intersection, if any of $e^x$ with $u x + v$. We examine the difference $e^x- ux -v$. It is a strictly convex function.

- if $u<0$, the function is strictly increasing and vanishes at a unique abscissa which will be the one of the your intersection point. Newton method whatever the starting point will converge to the root. Better to start on its right, I have not detailed more.
- if $u=0$ we have horizontal lines, left to reader.
- if $u>0$, this time the difference is still convex but with a minimum.  The value is $u - u \log u - v$. If this quantity is negative you have two intersection points, if it vanishes you have a tangent, if it is positive you have no intersection points.  To find the intersection points do Newton method with either large positive or large negative starting point.

Hope it helps.