Editing Natural logarithm (section)

{{Short description|Logarithm to the base of the mathematical constant e}}
{{Redirect|Base e|the numbering system which uses "e" as its base|Non-integer base of numeration#Base e}}
{{Use dmy dates|date=August 2019|cs1-dates=y}}

{{Infobox mathematical function
| name = Natural logarithm
| image = Log (2).svg
| imagesize = 290px
| imagealt = Graph of part of the natural logarithm function.
| caption = Graph of part of the natural logarithm function. The function slowly grows to positive infinity as {{mvar|x}} increases, and slowly goes to negative infinity as {{mvar|x}} approaches 0 ("slowly" as compared to any [[power law]] of {{mvar|x}}).
| general_definition = <math qid=Q204037>\ln x = \log_{e} x</math>
| motivation_of_creation = [[hyperbolic logarithm|hyperbola quadrature]]
| fields_of_application = Pure and applied mathematics
| domain = <math>\mathbb{R}_{> 0}</math>
| codomain = <math>\mathbb{R}</math>
| range = <math>\mathbb{R}</math>
| plusinf = +&infin;
| vr1 = {{mvar|e}}
| f1 = 1
| vr2 = 1
| f2 = 0
| vr3 = 0
| f3 = −∞
| asymptote = <math>x = 0</math>
| root = 1
| inverse = <math>\exp x</math>
| derivative = <math>\dfrac{d}{dx}\ln x = \dfrac{1}{x} , x > 0</math>
| antiderivative = <math>\int \ln x\,dx = x \left( \ln x - 1 \right) + C</math>
}}
{{E (mathematical constant)}}

The '''natural logarithm''' of a number is its [[logarithm]] to the [[base of a logarithm|base]] of the [[mathematical constant]] [[e (mathematical constant)|{{mvar|e}}]], which is an [[Irrational number|irrational]] and [[Transcendental number|transcendental]] number approximately equal to {{math|{{val|2.718281828459}}}}.<ref>{{Cite OEIS|A001113|Decimal expansion of e}}</ref> The natural logarithm of {{mvar|x}} is generally written as {{math|ln ''x''}}, {{math|log<sub>''e''</sub> ''x''}},  or sometimes, if the base {{mvar|e}} is implicit, simply {{math|log ''x''}}.<ref>G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers, 4th Ed., Oxford 1975, footnote to paragraph 1.7: "''log x is, of course, the 'Naperian' logarithm of x, to base e. 'Common' logarithms have no mathematical interest''".</ref><ref>{{cite book |title=Mathematics for physical chemistry |edition=3rd |author-first=Robert G. |author-last=Mortimer |publisher=[[Academic Press]] |date=2005 |isbn=0-12-508347-5 |page=9 |url=https://books.google.com/books?id=nGoSv5tmATsC}} [https://books.google.com/books?id=nGoSv5tmATsC&pg=PA9 Extract of page 9]</ref> [[Parentheses]] are sometimes added for clarity, giving {{math|ln(''x'')}}, {{math|log<sub>''e''</sub>(''x'')}}, or {{math|log(''x'')}}. This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity.

The natural logarithm of {{mvar|x}} is the [[exponentiation|power]] to which {{mvar|e}} would have to be raised to equal {{mvar|x}}. For example, {{math|ln 7.5}} is {{math|2.0149...}}, because {{math|1=''e''<sup>2.0149...</sup> = 7.5}}. The natural logarithm of {{mvar|e}} itself, {{math|ln ''e''}}, is {{math|1}}, because {{math|1=''e''<sup>1</sup> = ''e''}}, while the natural logarithm of {{math|1}} is {{math|0}}, since {{math|1=''e''<sup>0</sup> = 1}}.

The natural logarithm can be defined for any positive [[real number]] {{mvar|a}} as the [[Integral|area under the curve]] {{math|1=''y'' = 1/''x''}}  from {{math|1}} to {{mvar|a}}<ref name=":1">{{Cite web| last=Weisstein|first=Eric W.| title=Natural Logarithm|url=https://mathworld.wolfram.com/NaturalLogarithm.html| access-date=2020-08-29 | website=mathworld.wolfram.com | language=en}}</ref> (with the area being negative when {{math|0 < ''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 then be extended to give logarithm values for negative numbers and for all non-zero [[complex number]]s, although this leads to a [[multi-valued function]]: see [[complex logarithm]] for more.

The natural logarithm function, if considered as a [[real-valued function]] of a positive real variable, is the [[inverse function]] of the [[exponential function]], leading to the identities:
<math display="block">\begin{align}
   e^{\ln x} &= x \qquad \text{ if } x \in \R_{+}\\
   \ln e^x   &= x \qquad \text{ if } x \in \R
 \end{align}</math>

Like all logarithms, the natural logarithm maps multiplication of positive numbers into addition:<ref name=":2">{{cite web |title=Rules, Examples, & Formulas |department=Logarithm |url=https://www.britannica.com/science/logarithm|access-date=2020-08-29 |website=Encyclopedia Britannica |lang=en}}</ref>
<math display="block"> \ln( x \cdot y ) = \ln x + \ln y~.</math>

Logarithms can be defined for any positive base other than 1, not only {{mvar|e}}. However, logarithms in other bases differ only by a constant multiplier from the natural logarithm, and can be defined in terms of the latter, <math>\log_b x = \ln x / \ln b = \ln x \cdot \log_b e</math>.

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 scientific disciplines, and are used to solve problems involving [[compound interest]].