How to find function growth in this asymptotic analysis problem?

Question

Let $n=m!$ Which of the following is TRUE:

$m=\Theta \left( \frac{\log n}{log \log n} \right)m=\Theta \left( \frac{\log⁡n}{\log⁡\log⁡n} \right)$

$m=\Omega \left( \frac{\log n} {\log \log n} \right)$ but not $m=O \left( \frac{\log n} {\log \log n} \right)$

$m=\Theta(\log^2 n)$ $m=\Theta(\log^2 ⁡n)$

$m=\Omega(\log^2 n)$ but not $m=Ο(\log^2 ⁡n)$

$m=\Theta(\log^{1.5} n)$

Answer 1

If $n = m!$ then $\ln n =\ln m! =m\ln m - m+O(\ln m) =m\ln m(1+o(1)) $ so $\ln \ln n =\ln m +\ln \ln m+o(1) =\ln m(1+o(1)) $ so $\dfrac{\ln n}{\ln \ln n} =m(1+o(1)) $.