Logarithm
In mathematics, the logarithm of a number to a given base is the exponent to which the base must be raised in order to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 10 to the power of 3 is 1000: 103 = 1000. The logarithm of x to the base b is written logb(x) such as log10(1000) = 3.
By the following formulas, logarithms reduce multiplication to addition and exponentiation to products:
logb(x · y) | = | logb(x) + logb(y), |
logb(xp) | = | p logb(x). |
Three particular values for the base b are most common. The natural logarithm, the one with base b = e, occurs in calculus since its derivative is 1/x. The logarithm to base b = 10 is called common logarithm, while the base b = 2 gives rise to the binary logarithm.
The invention of logarithms is due to John Napier in the early 17th century. Before calculators became available, via logarithm tables, logarithms were crucial to simplifying scientific calculations. Today's applications of logarithms are numerous. Logarithmic scale reduces wide-ranging quantities to smaller scopes; this is applied in the Richter scale, for example. In addition to being a standard function used in various scientific formulas, logarithms appear in determining the complexity of algorithms and of fractals. They also form the mathematical backbone of musical intervals and some models in psychophysics, and have been used in forensic accounting. Logarithms are also closely related to questions revolving around counting prime numbers.
Different routines are available to calculate logarithms, some designed for speedy calculations, other ones for high accuracy. Logarithms have been generalized in various ways. The complex logarithm applies to complex numbers instead of real numbers. The discrete logarithm is an important primer in public-key cryptography.
Logarithm of positive real numbers
The logarithm of a number y with respect to a number b is the power to which b has to be raised in...