I wonder if it is widely taught, as far as heuristics go, that $\log N$ is just like $N^{1/\infty}$ (thinking something like $N^{0.00000000001}$). In other words, is there any danger in thinking this way?
I think a good test-case is deciding between $\log\log N$ and $\log^2 N := (\log N)^2$. (You may conclude that it doesn't help much.)