Nested Logarithm


In number theory (e.g., Ivić 2003), the symbol log_kx is commonly used to mean the nested logarithm (also called the repeated logarithm or iterated logarithm) ln...ln_()_(k)x, where lnx is the natural logarithm.

Care must be taken based on context as to when the notation log_kx denotes the logarithm to base k and when it means the k-nested natural logarithm.


The plots above show lnz, lnlnz, and lnlnlnz 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).

See also

Binary Logarithm, Common Logarithm, Logarithm, Natural Logarithm

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Canadian Mathematical Society. Student Mathematical Communicator. March 2001., J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, 2003.Ivić, A. "On a Problem of Erdős Involving the Largest Prime Factor of n." 5 Nov 2003.

Referenced on Wolfram|Alpha

Nested Logarithm

Cite this as:

Weisstein, Eric W. "Nested Logarithm." From MathWorld--A Wolfram Web Resource.

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