Do these functions have the same order of growth?

Question

I have a list of functions and was confused whether they have the same order of growth.

$$f(n) = \Theta(g(n))$$

Given functions: $\log^2 n, \log(n^2) $

My method: I took the logs of both functions and got an alternative form. which resulted in the following:

$\log((\log n)^2) \longrightarrow \log (2 \log n)$ and

$\log(\log (n^2)) \longrightarrow 2 \log \log n$

And through this was able to conclude that $ f (n) = \Theta(g(n))$ Is this true?

Answer 1

Hint: $\log(ab) = \log(a)+ \log(b)$