Comparing asymptotic order of logarithmic functions

Question

If I have two complicated logarithmic functions, say $\sqrt{\log n}$ and $\log(n(\log n)^3)$, and I have to compare them in terms of their asymptotic order. How do I do that? Do I have to create graphs, or is there another definitive way of doing that?

Answer 1

Using elementary properties of the logarithm we find out that $$\log(n \log^3 n) = \log n + 3\log\log n = O(\log n), $$ whereas $\sqrt{\log n} = o(\log n)$. So $\sqrt{\log n}$ grows more slowly.