Find an upper bound for $T(n)=T(\sqrt{n})+10\log\log n$
An alternative solution still using domain transformation/change of variables.

$$T(n) = T(\sqrt{n}) + \log \log n$$

1. Let $m = \log n$

We can then define a new function $S$ based on how $m$ changes in $T$ and expand it:

$$\begin{align}
S(m) &amp;= S\left(\frac{m}{2}\right) + \log m\\
&amp;=S\left(\frac{m}{4}\right) + \log m -1 + \log m\\
&amp; \vdots\\
&amp; = \log^2 m - \sum_{i = 1}^{\log m} i \\
&amp; = \Theta(\log^2 m) \implies T(n) = \Theta((\log \log n)^2) 
\end{align}$$

2. Let $k = \log \log n$

We can then define a new function $R$ based on how $k$ changes in $T$ and expand it:

$$\begin{align}
R(k) &amp;= R(k - 1) + k\\
&amp; = R(k - 2) + k - 1 + k\\
&amp; \vdots\\
&amp; = \Theta(k^2) \implies T(n) = \Theta((\log \log n)^2)
\end{align}$$