Iterated logarithm

In computer science, the iterated logarithm of

(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: In computer science, lg* is often used to indicate the binary iterated logarithm, which iterates the binary logarithm (with base

) instead of the natural logarithm (with base e).

Mathematically, the iterated logarithm is well defined for any base greater than

and base e. The "super-logarithm" function

is "essentially equivalent" to the base

iterated logarithm (although differing in minor details of rounding) and forms an inverse to the operation of tetration.

[2] 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:

the inverse grows much slower:

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.

Higher bases give smaller iterated logarithms.

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

In computational complexity theory, Santhanam[6] shows that the computational resources DTIME — computation time for a deterministic Turing machine — and NTIME — computation time for a non-deterministic Turing machine — are distinct up to