Prove or disprove: $(\ln n)^2 \in O(\ln(n^2)).$

Question

Prove or disprove: $(\ln n)^2 \in O(\ln(n^2)).$

I think I would start with expanding the left side. How would I go about this?

Answer 1

Here are two hints:

• $O(\ln(n^2)) = O(2 \ln n) = O(\ln n)$

• $x^2 \notin O(x)$

Can you finish from here?

Answer 2

Notice that: $$\forall n\in\mathbb{N}_{\geqslant 1},\ln\left(n^2\right)=2\ln(n).$$ Hence, one gets: $$O\left(\ln\left(n^2\right)\right)=O(\ln(n)).$$ Now, do you think $\ln(n)^2=O(\ln(n))$ i.e. is $(\ln(n))$ bounded?