Basic Examples

IteratedLog is the inverse of tetration (repeated exponentiation):
In[]:=
[◼]
IteratedLog
[E^E^E]
Out[]=
3
A slightly larger input shows a step-like jump in the value of
IteratedLog
:
In[]:=
[◼]
IteratedLog
[E^E^E+1]
Out[]=
4
Make a table of the iterated logarithm of the first 50 integers:
In[]:=
Table
[◼]
IteratedLog
[i],{i,0,50}
Out[]=
{0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}

Scope

The logarithmic base can be any real number greater than 1:
In[]:=
[◼]
IteratedLog
[Sqrt[2]+1/2,Pi^Pi^Pi]
Out[]=
5

Applications

Plot the iterated logarithm for different logarithmic bases:
In[]:=
ListLinePlot​​ Table
[◼]
IteratedLog
[i],{i,0,30},​​ Table
[◼]
IteratedLog
[8,i],{i,0,30},​​ Table
[◼]
IteratedLog
[20,i],{i,0,30}​​,PlotLegends{"iterated natural log","iterated log base-8","iterated log base-20"}
Out[]=
5
10
15
20
25
30
0.5
1.0
1.5
2.0
2.5
3.0
iterated natural log
iterated log base-8
iterated log base-20

Possible Issues

IteratedLog will return unevaluated in cases where evaluation might lead to numerical overflow:
In[]:=
[◼]
IteratedLog
[1.3,Pi^Pi^Pi]
Out[]=
[◼]
IteratedLog
1.3,
π
π
π
