I am having difficulty understanding how this follows.
$$(\log n)^{ (\log n) } = 2^{(\log n)(\log (\log n))} = n^{\log \log n}$$
Which logarithmic identities are used to go through each equality?
e.g. how do you first go from
$$(\log n)^{ (\log n) } = 2^{(\log n)(\log (\log n))}$$
and then to
$$2^{(\log n)(\log (\log n))} = n^{\log \log n}$$
(The log base must be 2 or else this equality won't hold)