What Logarithm rules says that someone can do this? $(\log n)^{\log n}$ is the same as $n^{\log \log n}$? What?
Just if anyone cares, for context, I am dealing with Time Complexities and I'm trying to figure out how they turned one such function $f(n) = (\log n)^{\log n}$ into this newly manipulated form.
You use the rules $e^{\log a} = a$ and $ \log (a^b) = b\log a$ to get $$ (\log n)^{\log n} = \exp(\log( (\log n)^{\log n})) = \exp(\log n \log\log n) = (\exp(\log n))^{\log \log n} = n^{\log\log n}. $$ [ Here $a = \log n$ and $b = \log n$ in the second formula above gives the second equality in the display.]