Is there a difference between $N\log{\log N}$ and $N\log^2N$

Question

I am trying to compare the growth rates of functions to review my understanding of basic Algorithms. The text asks to compare:

$$N\log \log N$$

and

$$N\log^2{N}$$

Are they not the same function?

Answer 1

Asking to compare $N\log(\log(N))$ with $N\log^2(N)$ would be useless if $\log^2(N)$ meant $\log(\log(N))$, therefore it most likely means $\log(N)^2$, i.e. $\log(N)\times\log(N)$, although I would avoid using that notation at all as being ambiguous. The first function grows slower.

Answer 2

$\log^2N$ is common notation for $(\log N)^2$ (compare $\sin^2x=(\sin x)^2$, $\cos^2x=(\cos x)^2$, etc. for example).

If we are talking about iterated logarithms, $\log_{j+1}N=\log(\log_jN)$ with $\log_1N=\log N$ is common (at least in analytic number theory). @StevenStadnicki also notes that a common alternative is $\log^{(j)}N$ to denote the $j$-fold iterated logarithm.