So given equations $log$ base $a$ of $x$ and $a$ to the $x$th power, I am asked to determine the values of $a$ for which the two graphs have one point of intersection and then the values of $a$ for which the two graphs have two points of intersection.
For the first problem, I know $0<a<1$ is an answer, but I feel like that there should also be an answer in the interval $a>1$, except that I don't quite know how to get that answer. And that also leaves me unable to answer question 2 as the answer would be in the form $1<a<$(some value where the two graphs intersect).
How would I solve this problem?
The trick here is to notice that since the two functions are inverses of each other, all the intersections lie on $y=x$. So you know the solutions satisfy the inverse of $x=a^x \implies a=x^{1/x}$, which has a max at $e^{1/e}$. Therefore, we have one intersection when $a\in(0,1)$, two when $a\in(1, e^{1/e})$, and zero when $a\in (\infty,0] \cup (e^{1/e}, \infty)$