Exponential Graph Intersections

The shown intersection $P$ lies to the left of 4. There, $x-4<0$, so $|x-4|=4-x$. Equal outputs give $1+e^{2x}=4-x$.

For $y=2+e^{-x}$, find the vertical-axis intercept and horizontal asymptote. Is the curve increasing or decreasing?

It decreases through (0,3) and approaches $y=2$ from above as $x$ increases.

The input $3-x$ must be positive, giving $x<3$. Do not write $\ln3-\ln x$: the subtraction is inside one log.

This is a rearrangement, not an explicit answer for $x$, because $x$ still appears on both sides. Numerical solution methods come later.