# 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 ${\displaystyle 1}$.[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, 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.

## Analysis of algorithms

The iterated logarithm is useful in analysis of algorithms and computational complexity, appearing in the time and space complexity bounds of some algorithms such as:

The iterated logarithm grows at an extremely slow rate, much slower than the logarithm itself, or repeats of it. This is because the tetration grows much faster than iterated exponential:

${\displaystyle {^{y}b}=\underbrace {b^{b^{\cdot ^{\cdot ^{b}}}}} _{y}\gg \underbrace {b^{b^{\cdot ^{\cdot ^{b^{y}}}}}} _{n}}$

the inverse grows much slower: ${\displaystyle \log _{b}^{*}x\ll \log _{b}^{n}x}$.

For all values of n relevant to counting the running times of algorithms implemented in practice (i.e., n ≤ 265536, which is far more than the estimated number of atoms in the known universe), the iterated logarithm with base 2 has a value no more than 5.

The base-2 iterated logarithm
x lg* x
(−∞, 1] 0
(1, 2] 1
(2, 4] 2
(4, 16] 3
(16, 65536] 4
(65536, 265536] 5

Higher bases give smaller iterated logarithms. Indeed, the only function commonly used in complexity theory that grows more slowly is the inverse Ackermann function [citation needed].

## Other applications

The iterated logarithm is closely related to the generalized logarithm function used in symmetric level-index arithmetic. The additive persistence of a number, the number of times someone must replace the number by the sum of its digits before reaching its digital root, is ${\displaystyle O(\log ^{*}n)}$.

In computational complexity theory, Santhanam[5] shows that the computational resources DTIMEcomputation time for a deterministic Turing machine — and NTIME — computation time for a non-deterministic Turing machine — are distinct up to ${\displaystyle n{\sqrt {\log ^{*}n}}.}$

## References

1. ^ Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2009) [1990]. "The iterated logarithm function, in Section 3.2: Standard notations and common functions". Introduction to Algorithms (3rd ed.). MIT Press and McGraw-Hill. pp. 58–59. ISBN 0-262-03384-4.
2. ^
3. ^ Alon, Noga; Azar, Yossi (April 1989). "Finding an approximate maximum" (PDF). SIAM Journal on Computing. 18 (2): 258–267. doi:10.1137/0218017. MR 0986665.
4. ^ Cole, Richard; Vishkin, Uzi (July 1986). "Deterministic coin tossing with applications to optimal parallel list ranking" (PDF). Information and Control. 70 (1): 32–53. doi:10.1016/S0019-9958(86)80023-7. MR 0853994.
5. ^