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Natural logarithm ：ウィキペディア英語版

Natural logarithm

The natural logarithm of a number is its logarithm to the base ''e'', where ''e'' is an irrational and transcendental constant approximately equal to . The natural logarithm of ''x'' is generally written as , , or sometimes, if the base ''e'' is implicit, simply .〔, (Extract of page 9 )
〕 Parentheses are sometimes added for clarity, giving ln(''x''), log_''e''(''x'') or log(''x''). This is done in particular when the argument to the logarithm is not a single symbol, to prevent ambiguity.
The natural logarithm of ''x'' is the power to which ''e'' would have to be raised to equal ''x''. For example, ln(7.5) is 2.0149..., because . The natural log of ''e'' itself, ln(''e''), is 1, because , while the natural logarithm of 1, ln(1), is 0, since .
The natural logarithm can be defined for any positive real number ''a'' as the area under the curve from 1 to ''a'' (the area being taken as negative when ''a''<1). The simplicity of this definition, which is matched in many other formulas involving the natural logarithm, leads to the term "natural". The definition of the natural logarithm can be extended to give logarithm values for negative numbers and for all non-zero complex numbers, although this leads to a multi-valued function: see Complex logarithm.
The natural logarithm function, if considered as a real-valued function of a real variable, is the inverse function of the exponential function, leading to the identities:
:

e^ = x \qquad \mboxx > 0

\ln(e^x) = x.

Like all logarithms, the natural logarithm maps multiplication into addition:
:

\ln(xy) = \ln(x) + \ln(y).

Thus, the logarithm function is a group isomorphism from positive real numbers under multiplication to the group of real numbers under addition, represented as a function:
:

\ln \colon \mathbb^+ \to \mathbb.

Logarithms can be defined to any positive base other than 1, not only ''e''. However, logarithms in other bases differ only by a constant multiplier from the natural logarithm, and are usually defined in terms of the latter. For instance, the binary logarithm is the natural logarithm divided by ln(2), the natural logarithm of 2. Logarithms are useful for solving equations in which the unknown appears as the exponent of some other quantity. For example, logarithms are used to solve for the half-life, decay constant, or unknown time in exponential decay problems. They are important in many branches of mathematics and the sciences and are used in finance to solve problems involving compound interest.

|-
|Indefinite Integral ||

x\ln x - x + C

|}
==History==
The concept of the natural logarithm was worked out by Gregoire de Saint-Vincent and Alphonse Antonio de Sarasa before 1649.〔R. P. Burn (2001) "Alphonse Antonio de Sarasa and Logarithms", Historia Mathematica 28:1 – 17〕 Their work involved quadrature of the hyperbola by determination of the area of hyperbolic sectors. Their solution generated the requisite "hyperbolic logarithm" function having properties now associated with the natural logarithm.
An early mention of the natural logarithm was by Nicholas Mercator in his work ''Logarithmotechnia'' published in 1668, although the mathematics teacher John Speidell had already in 1619 compiled a table of what in fact were effectively natural logarithms. It is also sometimes referred to as the Napierian logarithm, named after John Napier, although Napier's original "logarithms" (from which Speidell's numbers were derived) were slightly different (see Logarithm: from Napier to Euler).

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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