Proving Big-oh in both sides of equality

Question

I was searching for asymptotic notation examples to practice more. I came across this strange question:

Is it true of false that for any positive constants a and b, O(log_a(n)) = O(log_b(n))?

I've never seen big-oh notation in both sides! Can someone tell me how ?

Answer 1

I this case, we interpret $\mathcal O(f(n))$ as a set of functions; that is, you should think of $\mathcal O(f(n))$ as the set of all functions $g(n)$ for which you would otherwise write $g(n) = \mathcal O(f(n))$.

Hint: to prove that the two sets are equal, it suffices to show that $\log_a(n) = \mathcal O(\log_b(n))$ and $\log_b(n) = \mathcal O(\log_a(n))$.