Wolfram Research

Function Repository Resource:

IteratedLog

Source Notebook

Determine the iterated logarithm of an input

Contributed by: Wolfram|Alpha Math Team

ResourceFunction["IteratedLog"][z]

gives the iterated natural logarithm of z.

ResourceFunction["IteratedLog"][b,z]

gives the iterated logarithm base b of z.

Details and Options

The iterated logarithm is also known as inverse tetration or the super-logarithm. It is defined to be the smallest (integer) number of times that the logarithm must be applied to a number to yield a result less than 1.

Examples

Basic Examples

IteratedLog is the inverse of tetration (repeated exponentiation):

In[1]:=
ResourceFunction["IteratedLog"][E^E^E]
Out[1]=

A slightly larger input shows a step-like jump in the value of IteratedLog:

In[2]:=
ResourceFunction["IteratedLog"][E^E^E + 1]
Out[2]=

Make a table of the iterated logarithm of the first 50 integers:

In[3]:=
Table[ResourceFunction["IteratedLog"][i], {i, 0, 50}]
Out[3]=

Scope

The logarithmic base can be any real number greater than 1:

In[4]:=
ResourceFunction["IteratedLog"][Sqrt[2] + 1/2, Pi^Pi^Pi]
Out[4]=

Applications

Plot the iterated logarithm for different logarithmic bases:

In[5]:=
ListLinePlot[{
  	Table[ResourceFunction["IteratedLog"][i], {i, 0, 30}],
  	Table[ResourceFunction["IteratedLog"][8, i], {i, 0, 30}],
  	Table[ResourceFunction["IteratedLog"][20, i], {i, 0, 30}]
  }, PlotLegends -> {"iterated natural log", "iterated log base-8", "iterated log base-20"}]
Out[5]=

Possible Issues

IteratedLog will return unevaluated in cases where evaluation might lead to numerical overflow:

In[6]:=
ResourceFunction["IteratedLog"][1.3, Pi^Pi^Pi]
Out[6]=

Resource History

See Also

License Information