I was wondering how many times can we take the $\log$ of a given number. Does this number have a closed form? If so, what is it?
My try: I defined a function $f\colon\mathbb R^+\to\mathbb N$, which is our desired answer. Then, we get the following recursion and initial condition. \begin{gather} f(x)=f(\log x)+1\\ f(x)=1\forall x\in(0,1] \end{gather} Any ideas?