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 exact fraction can be given, as e is an irrational number just like pi). 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 Napierian logarithm.
The natural logarithm goes to minus infinity as x goes to 0.
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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) or, in the context of computing, log2(x). Sometimes also Log(x) (capital L) is used to mean log10(x), by those people who use log(x) with a lowercase l to mean loge(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.) As of 2005, some mathematicians have adopted the "ln" notation, but most use "log". In theoretical computer science, the base 2 logarithm is written as lg(x) to avoid confusion. This usage was suggested by Edward Reingold and popularized by Donald Knuth.
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
- <math>e^{\ln(x)} = x \,\!<math> for all positive x and
- <math>\ln(e^x) = x \,\! <math> for all real x.
In other words, the logarithm function is a bijection from the set of positive real numbers to the set of all real numbers. More precisely it is an isomorphism from the group of positive real numbers under multiplication to the group of real numbers under addition.
Logarithms can be defined to any positive base other than 1, 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 in a world using base 10 for nearly all calculations, this base 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 it 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{1}{x \cdot \ln b} <math>
When x is equal to 1, and the base (b) is e, then the slope of the graph will be 1.
There are other reasons the natural logarithm is natural: there are a number of simple series involving the natural logarithm, and it often arises in nature. Indeed, Nicholas Mercator first described them as log naturalis before calculus was even conceived.
Definitions
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:
- <math>\ln(ab)=\ln(a)+\ln(b) \,\!<math>
This can be shown by defining <math>\phi(t)=at<math> and using the substitution rule of integration as follows:
- <math>
\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 be defined as the unique real number a such that <math>\ln(a) = 1<math>.
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 <math>e^{\ln(x)} = x<math> 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
- <math>\frac{d}{dx}\ln(x)=\frac{1}{x}.<math>
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 <math>z = r e^{i \phi}<math> with r > 0 and <math>-\pi < \phi \le \pi<math>, then
- <math>\ln(z) = \ln(r) + i\phi \,\!<math>
So defined, ln is holomorphic for all complex numbers which are not non-positive reals, and it has the property
- <math>e^{\ln(z)} = z \,\! <math> 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 <math>z = r e^{i \phi}<math> we set
- <math>\ln(z) = \ln(r) + i(\phi + 2 \pi k) \,\!<math> : k any integer }
This is the set of all complex numbers u for which <math>e^u = z<math>, because <math>e^{2\pi i} = 1<math>(see Euler's identity).
The preferred way to deal with multivalued functions like this in complex analysis is via Riemann surfaces: the function ln is then not 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.
Numerical value
To calculate the numerical value of the natural logarithm of a number, the Taylor series expansion can be rewritten as:
- <math>\ln(1+x)= x \,( \frac{1}{1} - x\,(\frac{1}{2} - x \,(\frac{1}{3} - x \,(\frac{1}{4} - x \,(\frac{1}{5}- \ldots ))))) \quad{\rm for}\quad \left|x\right|<1.\,\!<math>
To obtain a better convergence, the following identity can be used.
- <math>\ln(x) = \ln(\frac{1+y}{1-y}) = 2\,y\, ( \frac{1}{1} + \frac{1}{3} y^{2} + \frac{1}{5} y^{4} + \frac{1}{7} y^{6} + \frac{1}{9} y^{8} + \ldots ) <math>
- <math>\ln(x) = \ln(\frac{1+y}{1-y}) = 2\,y\, ( \frac{1}{1} + y^{2} \, ( \frac{1}{3} + y^{2} \, ( \frac{1}{5} + y^{2} \, ( \frac{1}{7} + y^{2} \, ( \frac{1}{9} + \ldots ) ) ) ) ) <math>
provided that <math>y=\frac{x-1}{x+1}<math> and <math> 1 < x \,\!<math>
The natural logarithm in integration
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.
The natural logarithm can be integrated using integration by parts:
- <math>\int \ln (x) \,dx = x \ln (x) - x + C<math>
See also: Logarithmic integral function
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