Hi. I have a question for a computer science course. In the problem they manipulate an expression using logarithms.

They claim that $\displaystyle (\log{n})! = \theta((\log{n})^{\log{n} + \frac{1}{2}}e^{-\log(n)})$

By plugging $\displaystyle (\log{n})!$ into sterling's approximation $\displaystyle n! = \sqrt{2\pi n}(\frac{n}{e})^2$ and use the rules of logarithms to obtain the expression $\displaystyle (\log{n})^{\log{n} + \frac{1}{2}}e^{-\log(n)}$ which is just the result of plugging $\displaystyle (\log{n})!$ into sterling's approximation, manipulating the expression, and dropping lower order terms and constants(hence the theta notation). I'm not sure what they are doing to obtain these results. They claim that they apply the rule $\displaystyle a^{\log_b c} = c^{\log_b a}$ I know this is mostly algebraic so forgive me if this is the wrong forum but I figured it would be easier to explain away the missing parts of the final expression.