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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 products to sums and powers to products:

Three values for the base are used most often. The logarithm with base b = 10 is called the common logarithm; its primary use was for calculations before calculators could handle multiplication, division, powers, and roots effectively. The natural logarithm, with base b = e, occurs in calculus and is the inverse of the exponential function. The binary logarithm with base b = 2 has applications in computing.

Logarithms have a number of generalizations. The complex logarithm, the inverse of the exponential function applied to complex numbers, generalizes the logarithm to complex numbers. The discrete logarithm generalizes it to cyclic groups and has applications in public-key cryptography.

John Napier invented logarithms in the early 17th century, and since then they have been used for many applications in mathematics and science. Logarithm tables were used extensively to perform calculations, until replaced in the latter half of the 20th century by electronic calculators and computers. Logarithmic scales reduce wide-ranging quantities to smaller scopes; for example the Richter scale. They also form the mathematical backbone of musical intervals and some models in psychophysics, and have been used in forensic accounting. In addition to being a standard function used in various scientific formulas, logarithms are used in determining the complexity of algorithms and of fractals, and in prime-counting functions.


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