Growth rate of $a^x - b^x$ ($a > b > 1$) as $x \to \infty$
For any two real numbers $a$ and $b$ such that $a > b > 1$, we have that $\lim_{x\to\infty} a^x - b^x = \infty$. What can we say about the growth rate of $a^x - b^x$? (For instance, can it be described as $c^x(1+o(1))$ for some $c$? I'm pretty sure it's superpolynomial, at least, but I have no idea how to prove that.)
Since we can explicitly write $$ a^x - b^x = a^x \left[ 1 - \left(\frac{b}{a}\right)^x \right] $$ and $(b/a)^x = o(1)$, this puts the function in the form desired, with $c = a$.