Iterated logarithm

In computer science, the iterated logarithm of ${\displaystyle n}$, written log* ${\displaystyle n}$ (usually read "log star"), is the number of times the logarithm function must be iteratively applied before the result is less than or equal to ${\displaystyle 1}$. The simplest formal definition is the result of this recurrence relation:

${\displaystyle \log ^{*}n:={\begin{cases}0&{\mbox{if }}n\leq 1;\\1+\log ^{*}(\log n)&{\mbox{if }}n>1\end{cases}}}$

On the positive real numbers, the continuous super-logarithm (inverse tetration) is essentially equivalent:

${\displaystyle \log ^{*}n=\lceil \mathrm {slog} _{e}(n)\rceil }$

i.e. the base b iterated logarithm is ${\displaystyle \log ^{*}n=y}$ if n lies within the interval ${\displaystyle ^{y-1}b<n\leq \ ^{y}b}$, where ${\displaystyle {^{y}b}=\underbrace {b^{b^{\cdot ^{\cdot ^{b}}}}} _{y}}$denotes tetration. However, on the negative real numbers, log-star is ${\displaystyle 0}$, whereas ${\displaystyle \lceil {\text{slog}}_{e}(-x)\rceil =-1}$ for positive ${\displaystyle x}$, so the two functions differ for negative arguments.

The iterated logarithm accepts any positive real number and yields an integer. Graphically, it can be understood as the number of "zig-zags" needed in Figure 1 to reach the interval ${\displaystyle [0,1]}$ on the x-axis.

In computer science, lg* is often used to indicate the binary iterated logarithm, which iterates the binary logarithm (with base ${\displaystyle 2}$) instead of the natural logarithm (with base e).

Mathematically, the iterated logarithm is well-defined for any base greater than ${\displaystyle e^{1/e}\approx 1.444667}$, not only for base ${\displaystyle 2}$ and base e.