How to solve T(n)=2T(√n)+log n with the master theorem?
Let us actually use the master theorem.

Define $S(n) = T(e^n)$ for all $n$. Then
  $$S(n) = T(e^n) = 2T(\sqrt{e^n}) + \log(e^n) = 2T(e^{n/2}) + n = 2S(n/2) + n$$
Now we can apply the second case of the master theorem to $S(n)$ for $a = b = 2$ and $f(n) = n$ to obtain
  $$ S(n) = \Theta(n\log n)$$
So for $n\gt0$,
  $$ T(n) = S(\log n) = \Theta(\log n \log\log n)$$