Asymptotic Complexity of Logarithms and Powers

Question

So, clearly, log(n) is O(n). But, what about (log(n))^2? What about sqrt(n) or log(n)--what bounds what?

There\'s a family of comparisons like this:
n^a versu

Answer 1

log n -- O(log n)
sqrt n -- O(sqrt n)
n^2 -- O(n^2)
(log n)^2 -- O((log n)^2)

n^a versus (log(n))^b

You need either bases or powers the same. So use your math to change n^a to log(n)^(whatever it gets to get this base) or (whatever it gets to get this power)^b. There is no general case