$a)$ Determine for all pairs $i$ and $j$, $i,j ∈ {1, ldots, 6}$ whether for the ones given below functions $f_i ∈ O(f_j)$ or $f_i ∈ o(f_j)$ or neither of the two applies as $n → infty$:

$f_1 = log(n),$

$f_2 = log(sqrt n),$

$f_3 = log(n + log(n)),$

$f_4 = sqrt{log(n)},$

$f_5 = loglog(n^{log(n)}),$

$f_6 = log_2(n)$

$b)$ Show that for the functions $ f(n) = log(n!) $ and $g(n) = nlog(n)$ it holds that

$f ∈ O(g)$ and $g ∈ O(f)$. In fact, $f$ and $g$ are even asymptotically equivalent.

I am confused about the big $O$ and small $o$. When is $f_{i} ∈ O(f_j)$ and when is $f_i ∈ o(f_j)$? How to prove asymptotic equivalence?