Logarithmic growth

In mathematics, logarithmic growth describes a phenomenon whose size or cost can be described as a logarithm function of some input. e.g. y = C log (x). Note that any logarithm base can be used, since one can be converted to another by a fixed constant. Logarithmic growth is the inverse of exponential growth and is very slow.

A familiar example of logarithmic growth is the number of digits needed to represent a number, N, in positional notation, which grows as log_b (N), where b is the base of the number system used, e.g. 10 for decimal arithmetic. Another example is in cryptography, where the key size needed to protect against a brute force attack for a certain period of time grows logarithmically with the desired protection interval.

In the design of computer algorithms, logarithmic growth, and related variants, such as log-linear, or linearithmic, growth are very desirable indications of efficiency.

Logarithmic growth can lead to apparent paradoxes, as in the martingale roulette system, where the potential winnings before bankruptcy grow as the logarithm of the gambler's bankroll. It also plays a role in the St. Petersburg paradox.

In microbiology, the rapidly growing exponential growth phase of a cell culture is sometimes called logarithmic growth. During this bacterial growth phase, the number of new cells appearing are proportional to the population.

See also

• Iterated logarithm - an even slower growth model

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