# Iterated logarithm

In computer science, the iterated logarithm of $n$ , written log* $n$ (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 $1$ . The simplest formal definition is the result of this recurrence relation:

$\log ^{*}n:={\begin{cases}0&{\mbox{if }}n\leq 1;\\1+\log ^{*}(\log n)&{\mbox{if }}n>1\end{cases}}$ On the positive real numbers, the continuous super-logarithm (inverse tetration) is essentially equivalent:

$\log ^{*}n=\lceil \mathrm {slog} _{e}(n)\rceil$ i.e. the base b iterated logarithm is $\log ^{*}n=y$ if n lies within the interval $^{y-1}b , where ${^{y}b}=\underbrace {b^{b^{\cdot ^{\cdot ^{b}}}}} _{y}$ denotes tetration. However, on the negative real numbers, log-star is $0$ , whereas $\lceil {\text{slog}}_{e}(-x)\rceil =-1$ for positive $x$ , so the two functions differ for negative arguments. Figure 1. Demonstrating log* 4 = 2 for the base-e iterated logarithm. The value of the iterated logarithm can be found by "zig-zagging" on the curve y = logb(x) from the input n, to the interval [0,1]. In this case, b = e. The zig-zagging entails starting from the point (n, 0) and iteratively moving to (n, logb(n) ), to (0, logb(n) ), to (logb(n), 0 ).

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 $[0,1]$ on the x-axis.

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

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

## 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 $O(\log ^{*}n)$ .

In computational complexity theory, Santhanam 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 $n{\sqrt {\log ^{*}n}}.$ 