Nested Logarithm

A nested logarithm is a fixed-depth function composition of logarithms. In number theory (e.g., Ivić 2005), the symbol is commonly used for the -fold nested natural logarithm , which is also called a repeated logarithm and, in this context, an iterated logarithm.

Care must be taken based on context as to when the notation denotes the logarithm to base and when it means the -nested natural logarithm. In the analysis of algorithms, the iterated logarithm usually means the log-star function , which counts iterations rather than evaluating a fixed-depth composition.

The plots above show , , and in the complex plane.

The penchant for formulas and bounds containing a profusion of nested logarithms has led to the following joke. What sound does a drowning analytic number theorist make? A: log log log log... (Havil 2003, p. 115).