Correct order of the growth function

Question

$5 \log( \log n) $

$n (\log n)^2$

$\sqrt{n} \log n$

$n^{\frac{4}{3}}$

$n \log (\log n)$

$7 \sqrt{n}$

What is the ascending order of the growth function? Please give the explanation as well.

Answer 1

If you take sufficiently large value of n,then ascending order according to the value will be, $5 log( log n) < 7 \sqrt{n}<\sqrt{n} log n<n log (log n)<n (log n)^2<n^{\frac{4}{3}}$

So,if we sort them from Most efficient to least efficient,it will be

$5 log( log n) > 7 \sqrt{n}>\sqrt{n} log n>n log (log n)>n (log n)^2>n^{\frac{4}{3}}$

Answer 2

If we want for sufficient large n, then if we name the equation from top to down as from 1 to 6 then ordering looks like 1<5<2 , 7<3<2 and 6<4.