In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, logarithmic growth describes a phenomenon whose size or cost can be described as a
logarithm
In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is , because is to the rd power: . More generally, if , the ...
function of some input. e.g. ''y'' = ''C'' log (''x''). Any logarithm base can be used, since one can be converted to another by multiplying by a fixed constant.
[.] Logarithmic growth is the inverse of
exponential growth
Exponential growth occurs when a quantity grows as an exponential function of time. The quantity grows at a rate directly proportional to its present size. For example, when it is 3 times as big as it is now, it will be growing 3 times as fast ...
and is very slow.
A familiar example of logarithmic growth is a number, ''N'', in
positional notation
Positional notation, also known as place-value notation, positional numeral system, or simply place value, usually denotes the extension to any radix, base of the Hindu–Arabic numeral system (or decimal, decimal system). More generally, a posit ...
, which grows as log
''b'' (''N''), where ''b'' is the base of the number system used, e.g. 10 for decimal arithmetic. In more advanced mathematics, the
partial sums of the
harmonic series
:
grow logarithmically. In the design of computer
algorithm
In mathematics and computer science, an algorithm () is a finite sequence of Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algo ...
s, logarithmic growth, and related variants, such as log-linear, or
linearithmic, growth are very desirable indications of efficiency, and occur in the
time complexity
In theoretical computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations ...
analysis of algorithms such as
binary search.
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
Microbiology () is the branches of science, scientific study of microorganisms, those being of unicellular organism, unicellular (single-celled), multicellular organism, multicellular (consisting of complex cells), or non-cellular life, acellula ...
, 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 is proportional to the population. This terminological confusion between logarithmic growth and exponential growth may be explained by the fact that exponential growth curves may be straightened by plotting them using a
logarithmic scale
A logarithmic scale (or log scale) is a method used to display numerical data that spans a broad range of values, especially when there are significant differences among the magnitudes of the numbers involved.
Unlike a linear Scale (measurement) ...
for the growth axis.
[.]
See also
* (an even slower growth model)
References
{{DEFAULTSORT:Logarithmic Growth
Logarithms