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Natural logarithm

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The natural logarithm is the logarithm to the base e, where e is approximately equal to 2.71828... (no precise decimal fraction can be given, as e is an irrational number). The natural logarithm is defined for all positive real numbers x and can also be defined for non-zero complex numbers as will be explained below. Although this function was not introduced by Napier, it is sometimes known as the Naperian Logarithm.


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Notational conventions

Mathematicians generally understand either "ln(x)" or "log(x)" to mean loge(x), i.e., the natural logarithm of x, and write "log10(x)" if the base-10 logarithm of x is intended. Engineers, biologists, and some others write only "ln(x)" or (occasionally) "loge(x)" when they mean the natural logarithm of x, and take "log(x)" to mean log10(x).

Most of the reason for thinking about base-10 logarithms became obsolete shortly after about 1970 when hand-held calculators became widespread (for more on this point, see common logarithm). Nonetheless, since calculators are made and often used by engineers, the conventions to which engineers were accustomed continued to be used on calculators, so now most non-mathematicians take "log(x)" to mean the base-10 logarithm of x and use only "ln(x)" to refer to the natural logarithm of x. As recently as 1984, Paul Halmos[?] in his autobiography heaped contempt on what he considered the childish "ln" notation, which he said no mathematician had ever used. (The notation was in fact invented in 1893 by Irving Stringham, professor of mathematics at Berkeley.) At the time of this writing (2003), many mathematicians have adopted the "ln" notation, but "log" also remains in widespread use.

To avoid all confusion, Wikipedia uses the notation ln(x) for the natural logarithm of x and log10(x) for the base-10 logarithm of x.

Ln is the inverse of the natural exponential function

This function is the inverse function of the exponential function, thus it holds

eln(x) = x      for all positive x and
ln(ex) = x      for all real x.

Logarithms can be defined to any base, not just e, and they are always useful for solving equations in which the unknown appears as the exponent of some other quantity.

What's so "natural" about them?

Initially, it seems that the base-10 would be more "natural" than base e. The reason we call ln(x) "natural" is twofold: first, the natural logarithm can be defined quite easily using a simple integral or Taylor series as will be explained below; this is not true of other logarithms. Second, expressions in which the unknown variable appears as the exponent of e occur much more often than exponents of 10 (because of the "natural" properties of the exponential function which allow to describe growth and decay behaviors), and so the natural logarithm is more useful in practice. To put it concretely, consider the problem of differentiating a logarithmic function:

<math>\frac{d}{dx}\log_b(x)=\frac{[{\rm constant}]}{x}.</math>
Only when the base of the logarithmic function is e is the "constant" equal to 1.


Formally, ln(a) may be defined as the area under the graph (integral) of 1/x from 1 to a, that is,

<math>\ln(a)=\int_1^a \frac{1}{x}\,dx.</math>

This defines a logarithm because it satisfies the fundamental property of a logarithm:

This can be shown by defining &phi(t)=at and using the substitution rule of integration as follows:

\ln (ab)

\int_1^{ab} \frac{1}{x} \; dx

\int_1^a \frac{1}{x} \; dx \; + \int_a^{ab} \frac{1}{x} \; dx

\int_1^{a} \frac{1}{x} \; dx \; + \int_1^{b} \frac{1}{t} \; dt

\ln (a) + \ln (b) </math>

The number e can then defined as the unique real number with ln(e) = 1.

Alternatively, if the exponential function has been defined first using an infinite series, the natural logarithm may be defined as its inverse function, meaning ln(x) is that number for which eln(x) = x. Since the range of the exponential function is all positive real numbers and since the exponential function is strictly increasing, this is well-defined for all positive x.

Derivative, Taylor series and complex arguments

The derivative of the natural logarithm is given by

This leads to the Taylor series
<math>\ln(1+x)=\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} x^n\quad{\rm for}\quad \left|x\right|<1.</math>

One may define ln(z) also for all non-zero complex numbers z. The above Taylor expansion remains valid for all complex numbers x with absolute value less than 1. If the non-zero complex number z is expressed in polar coordinates as z = r eiφ with r > 0 and -π < φ ≤ +π, then

ln(z) = ln(r) + iφ
So defined, ln is holomorphic for all complex numbers which are not non-positive reals, and it has the property
eln(z) = z     for all nonzero z
One has to be careful, because several properties familiar from the real logarithm are no longer valid for this complex extension. For example, ln(ez) does not always equal z, and ln(zw) does not always equal ln(z) + ln(w).

A somewhat more natural definition of ln(z) interprets it as a multi-valued function[?]: for z = r eiφ we set

ln(z) = { ln(r) + i(φ + 2πk) : k any integer }
This is the set of all complex numbers u for which eu = z, because ei = 1 (see The most remarkable formula in the world).

The preferred way to deal with multivalued functions like this in complex analysis is via Riemann surfaces: the function ln is then non defined on the complex plane but instead on a suitable Riemann surface having countably many "leaves" and the values of the function differ by 2πi from leaf to leaf.

Ln integration condition

The natural logarithm allows simple integration of functions of the form g(x) = f '(x)/f(x): an antiderivative of g(x) is given by ln(|f(x)|). This is the case because of the chain rule and the following fact:

<math>{d \over dx}\left( \ln \left| x \right| \right) = {1 \over x}</math>

Here is an example in the case of g(x) = tan(x):

<math>\int \tan (x) \,dx = \int {\sin (x) \over \cos (x)} \,dx</math>
<math>\int \tan (x) \,dx = \int {-{d \over dx} \cos (x) \over {\cos (x)}} \,dx</math>
Letting f(x) = cos(x) and f'(x)= - sin(x):
<math>\int \tan (x) \,dx = -\ln{\left| \cos (x) \right|} + C</math>
<math>\int \tan (x) \,dx = \ln{\left| \sec (x) \right|} + C</math>

where C is an arbitrary constant of integration.

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