ITERATED LOGARITHM Meaning and
Definition
-
The iterated logarithm is a mathematical function used to measure the growth rate of a logarithm, specifically when applying repeated logarithmic operations. It is denoted as log* n and represents the number of times the logarithm function (base 2) must be repeated to reduce a positive integer n to a value less than or equal to 1.
More formally, the iterated logarithm of n is defined as follows: log* n = min{ k ≥ 0 : log(log(...log n...)) ≤ 1 }, where the logarithm operation is repeated k times.
The iterated logarithm function is primarily used in the analysis of algorithms and computational complexity theory. It provides a valuable tool to analyze algorithms with very slow growth rates. For example, it can quantify the running time of algorithms with logarithmic complexities such as binary search trees or divide-and-conquer algorithms.
The value of the iterated logarithm grows very slowly; it increases extremely slowly with the size of the input. For practical purposes, it is often considered a constant, typically less than or equal to 5, regardless of the input size.
Overall, the iterated logarithm function serves as a tool to express the efficiency of algorithms and determine their worst-case behavior when dealing with problems of logarithmic complexity.
Etymology of ITERATED LOGARITHM
The word "iterated logarithm" is derived from the combination of two concepts: "iteration" and "logarithm".
The term "iteration" refers to the process of repeating a sequence of steps or operations multiple times. It comes from the Latin word "iterare", which means "to repeat" or "to do again".
On the other hand, "logarithm" is a mathematical function used to determine the power to which a base must be raised to obtain a given number. The word "logarithm" originated from two Greek words: "logos", meaning "word" or "ratio", and "arithmos", meaning "number".
When these two terms are combined, "iterated logarithm" describes the repeated application of the logarithm function. It represents the concept of taking the logarithm of a number, then taking the logarithm of that result, and so on for multiple iterations.
