What is Big-O hierarchy

Question

The big-O hierarchy: A few basic facts about the big-O behaviour of some familiar functions are very important. Let p(n) be a polynomial in n (of any degree). Then

log_bn is O(p(n)) and p(n) is O(aⁿ);

for any base b and any a. In words: logs are big-O of polynomials and polynomials are big-O of exponentials.

Note that since log_bn = log_cn/ log_cb, we have

log_bn is O(log_cn);

for any fixed b and c, since log_cb is a constant.