Editing Iterated logarithm (section)

{{Short description|Inverse function to a tower of powers}}
{{For|the theorem in probability theory|Law of the iterated logarithm}}
[[Image:Iterated logarithm.png|right|300px|thumb|'''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 = log<sub>b</sub>(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, log<sub>b</sub>(n) ), to (0, log<sub>b</sub>(n) ), to (log<sub>b</sub>(n), 0 ).]]
In [[computer science]], the '''iterated logarithm''' of <math>n</math>, written {{log-star}}&nbsp;<math>n</math> (usually read "'''log star'''"), is the number of times the [[logarithm]] function must be  [[iteration|iteratively]] applied before the result is less than or equal to <math>1</math>.<ref>{{Introduction to Algorithms|3|chapter=The iterated logarithm function, in Section 3.2: Standard notations and common functions|pages=58–59}}</ref> The simplest formal definition is the result of this [[recurrence relation]]:

:<math>
  \log^* n :=
  \begin{cases}
    0                  & \mbox{if } n \le 1; \\
    1 + \log^*(\log n) & \mbox{if } n > 1
   \end{cases}
 </math>

In computer science, '''{{lg-star}}''' is often used to indicate the '''binary iterated logarithm''', which iterates the [[binary logarithm]] (with base <math>2</math>) instead of the natural logarithm (with base ''e''). Mathematically, the iterated logarithm is well defined for any base greater than <math>e^{1/e} \approx 1.444667</math>, not only for base <math>2</math> and base ''e''. The "super-logarithm" function <math>\mathrm {slog}_b(n)</math> is "essentially equivalent" to the base <math>b</math> iterated logarithm (although differing in minor details of [[rounding]]) and forms an inverse to the operation of [[tetration]].<ref>{{cite journal
 | last1 = Furuya | first1 = Isamu
 | last2 = Kida | first2 = Takuya
 | doi = 10.3390/a12080159
 | issue = 8
 | journal = Algorithms
 | mr = 3998658
 | article-number = 159
 | title = Compaction of Church numerals
 | volume = 12
 | year = 2019| page = 159
 | doi-access = free
 }}</ref>