Change of variables in function $T(n)$.

Question

I've been given this recurrence to solve:

$T(n) = T(\sqrt n) + \theta(lglgn)$

And I'm told that the way to solve it is to let $m = lgn$, so that the recurrence can be rewritten as follows:

$S(m) = S(m/2) + \theta(lgm)$

But I don't understand how $T(\sqrt n)$ can become $S(m/2)$. Is there some key manipulation going on that I'm not seeing?

Answer 1

$$ \frac{1}{2} m = \frac{1}{2} \log n = \log n^\frac{1}{2} = \log \sqrt{n}$$ $$ S(\log n) = S(\log \sqrt{n}) + \Theta(\log \log n) $$