Unique intersection of $b^x$ and $\log_b(x)$

Question

It seems to me that there is exactly one real number $b>1$ such that the graphs of $y=b^x$ and $y=\log_b(x)$ intersect at a single point. What exactly is this number?

Answer 1

The two functions are inverse one of the other. When they intersect in inly one point, they do it in a point where the slope of the graph of the functions is $1$. To find $b$ solve the equations $$ b^x=\log_bx,\quad(\log b)\,b^x=\frac{1}{(\log b)\,x}=1. $$ The solution is $b=e^{1/e}$, $x=e$.