What is the big-o of the function 2 log(log n) + 3 n log(n) + 5 log(n)?
Starting with an analytical assessment of the function's terms, we'll first format all terms as similarly as possible to each other, in order to more easily compare with one another. To do so, all constants will be removed, leaving only the variable elements of each term. This is acceptable to do since we're ultimately seeking Big O classification, which typically sets all constants within its expression to a value of $1$.


$2log(log(n)) \geq log(log(n))$
$3n\: log(n) \geq n\: log(n)$
$5log(n) \geq log(n)$


Rewriting, 

$\quad f(n) = log(log(n)) + n\: log(n) + log(n)$,

where

$\quad n \in \mathbb{N}$

Now, since each term varies only by their logarithmic arguments, we can establish an inequality by direct comparison of those arguments, resulting in

$\quad log(log(n)) \lt log(n) \lt n\: log(n), \forall n \gt 1$ 

Therefore, it follows that 

$\quad f(n) \in O(n\:log(n))$