I was helping a graduate student with a problem, and I realized that I didn't have a good intuitive explanation for the following phenomenon:
$$ \int_e^\infty \frac{dx}{x\ln^p(x)}<\infty \quad \forall p>1 $$
Obviously the result follows by integration with a $u=\ln(x)$ substitution. However, I found it difficult to contrast this with the fact that $\ln^p(x)$ grows slower than any positive power of $x$, which "intuitively" suggests that the logarithmic factor won't improve the integrability of $1/x$ by itself. Can anyone think of an intuitive (i.e. non-rigorous, "formal", whatever) way to explain this?