If $a > b > 0$ then $\lim_{x\to \infty} \frac {a^x}{b^x} = \infty $
So we $a^x$ grows faster than $b^x$
Doing the same thing for their inverses yields: $\lim_{x\to \infty} \frac {\log_ax}{\log_bx} = \frac {\log b}{\log a} \neq 0$
Intuitively since $a^x$ grows faster than $b^x$, one would assume that the inverse of $a^x$ would grow slower that the inverse of $b^x$. After all their graphs are just the mirror images across $y = x$. It appears that algebraicaly we do not reach that conclusion? Is there an intuitive explanation of this?