The number of logarithm applications to get from n below 1

Question

Let $L(n)$ to be a number of logarithms that you need to apply on $n$ until you get below 1:

$$ 0 \leq \log\cdots\log n < 1 \\ \uparrow \\ L(n)\mbox{-times} $$

Is there a name for this function? It's clearly very very slowly growing. Is it perhaps somehow related to the inverse Ackermann function?

Answer 1

It is known as the iterated logarithm function $\log^{*}(n)$, with a small modification. Note that we have that $\log^{*}(n)=0$ if $0 \leq n \leq 1$. However your $L(1)=1$. This will cause that $L(n) \neq \log^{*}(n)$ if $n$ is in the following sequence:

$$1,e,e^e, e^{e^e}, e^{e^{e^e}}, e^{e^{e^{e^e}}}, \cdots$$

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The inverse Ackermann function $\alpha(n)$ is even slower. For example, consider the powertower $b=e^{e^{e^{e^{\dots}}}}$ with height $10^9$. Then $\log^{*}(n)=10^9+1$, $L(n)=10^9$ but $\alpha(n)=5$.

Answer 2

It grows very fast compared to the Ackermann function.

Indeed,

$$L( e \uparrow \uparrow n ) = n$$

Where $\uparrow$ is the Knuth's up arrow