Comparing algorithm running times expressed in complex form

Question

I know how to compare running times of different algorithms.

Sometimes it is obvious, sometimes it requires simplifications, and sometimes dividing and using L'Hopital's rule to see if it converges to a constant or 0.

However, how do you compare the functions if they are in a complex form?

For example, how would you compare

and n?

Of course, I want a rigorous proof.

Answer 1

Take logs on both side, i.e. compare $(\log_{96} n)^{\sqrt 2-.4}$ with $\log_2 n$, simplify the former to $\left(\frac{\log_2 n}{\log_2 06}\right)^{\sqrt 2-.4}=c\cdot(\log_2n)^{\sqrt 2-.4}$, so you are essentially comparing $x$ with $x^{\sqrt2-.4}$. Since $\sqrt 2-.4>1$, the latter wins.