Exponential and Logarithmic Graphs

A function’s domain is its allowed inputs; its range is its possible outputs. Since $e^x$ takes any real input and gives a positive output, $\ln x$ takes positive inputs and gives any real output.

If $(u,v)$ lies on $y=e^x$, then $v=e^u$ and $u=\ln v$. Thus $(v,u)$ lies on $y=\ln x$. Swapping coordinates reflects a graph in $y=x$.

An asymptote is a straight line the curve approaches arbitrarily closely in a stated direction. Here $e^x$ approaches $y=0$ as $x$ decreases without limit. It never reaches zero. The inverse curve approaches $x=0$ from the right as its output decreases without limit.

Sketch $y=e^x$ and $y=\ln x$. Label their axis intercepts and asymptotes. What point on the second corresponds to $(-1,1/e)$ on the first?

The exponential crosses the vertical axis at (0,1), with asymptote $y=0$. The log crosses the horizontal axis at (1,0), with asymptote $x=0$. Neither crosses its asymptote.