So I have a question for the recurrence $T(n) = 2T(\sqrt{n}) + \log n$. We are to use substitution method to figure out the solution. This is an example problem (not a exercise problem) in my book (Introduction to Algorithms, Third Edition pg. 86). I'm having a hard time understanding how they rename $m = \log n$.
They then get the new equation $T(2^m) = 2T(2^{m/2}) + m$. I see that turning the $m = \log n$ into $2^m = 2^{\log n}$ and using the property $a^{\log b} = b^{\log a}$ so you get $2^m = n^{\log 2}$ which goes to $2^m = n$ and hence $\sqrt{n} = 2^{m/2}$.
My question is, how do you know to guess to substitute $m$ for $\log n$?