One example in my textbook says that
$$\log\log n^n=\log n\log n=\log n+\log\log n = \mathcal{O}(\max(\log n, \log\log n))=\mathcal{O}(\log\log n)$$
and I was wondering why is that true? I thought that
$$\log n \leq n \Leftrightarrow \log\log n \leq \log n.$$
So, for $C=1$ and $n\geq 1$ we have $\log\log n =\mathcal{O}(\log n)$.
Thanks for any advice.
