Why does $\log x$ go to $-\infty$ so slowly? While checking convergence of integrals, I always have to remind myself that $\log x$ goes to infinity very slowly. This is almost like and axiom to me. I don't intuitively get why it goes to infinity so slowly. On a graph, it sure doesn't look that way. Any thoughts?
More precisely:
$$\forall\delta>0: \lim_0x^\delta\log x=0$$

Because $\log(x^{\delta})=\delta\log(x)$, which is essentially $\log(x)$, and
$$\log(\log(x))\leftrightarrow \log(x)\leftrightarrow x\leftrightarrow e^x\leftrightarrow e^{e^x}$$
truly represent different growth behaviors.
