# Math Help - Size of functions

1. ## Size of functions

Hello Everyone!

I have a couple of questions:

(1) is $(\log n!)^3 = O(3n^3+3)$ true?

(2) is $n^{\log _2 n} = O(2^{\sqrt{n}})$ true?

I'm going for yes for (2), but have no clue regarding (1)

2. For the first one,
if $(\log n!)^3 \in O(3n^3+3) \subseteq O(n^3)$
then $(\log n!)^3 \leq C^3n^3$ for some $C$ and $n$ sufficiently large.
So $\log n! \leq Cn$ or $n! \leq a^{n}$ which is not true for $n$ much larger than $a$. Contradiction.

For the second one,
$n^{\log _2 n} = 2^{(\log_2 n)^2}$
Now show that $(\log_2 n)^2 \in O(\sqrt{n})$.