I was going across some problems which translated the fact that $\log (n!) = \Theta(n\log n)$ to give $$\log(\lceil\log n\rceil!) = \Theta(\lceil\log n\rceil \log \lceil\log n\rceil)$$ and therefore $\log(\lceil \log n\rceil!) = \omega(\lg n)$.
Now I just have this intuition that this sort of conversions shouldn't and wouldn't be always right.
Would it be right to think that.
Any justification as to why am I wrong in thinking that and guidance to what I should have a look at, what texts I should go through to understand it better for myself are very much appreciated.