Natural logarithm

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The natural logarithm is the logarithm to the base e, where e is equal to 2.718281828459... (continuing forever). 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.

Contents 1 Notational conventions 2 Ln is the inverse of the natural exponential function 3 Reason for being "natural" 4 Definitions 5 Derivative, Taylor series and complex arguments 6 Numerical value 7 The natural logarithm in integration 8 See also 9 References

Notational conventions

• Mathematicians generally understand either "ln(x)" or "log(x)" to mean log_e(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) "log_e(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).

• In most commonly-used programming languages, including C, [[C++]], Fortran, and BASIC, "log" or "LOG" means natural logarithm.

• On hand-held calculators the natural logarithm is ln, whereas log is the base-10 logarithm.

See also logarithms.

Ln is the inverse of the natural exponential function

This function is the inverse function of the exponential function:

<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.

Reason for being "natural"

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, 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); second, the natural logarithm can be defined quite easily using a simple integral or Taylor series as will be explained below, and this is not true of other logarithms. Thus, 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>

If the base (b) is e then the derivative is 1/x and at x=1 the slope of the graph is 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 = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots \quad{\rm for}\quad \left|x\right|<1,</math>

which is also known as the Mercator series.

An alternative form <ref>"Logarithmic Expansions" at Math2.org</ref> for ln(x) itself is

<math>\ ln(x)=\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} (x-1) ^ n = x - 1 - \frac{(x-1) ^ 2}{2} + \frac{(x-1)^3}{3} - \frac{(x-1)^4}{4} \cdots \quad{\rm for}\quad 0<x<2,</math>

One may define ln(z) also for all non-zero complex numbers z, but it is usually denoted log(z) for mostly two reasons: To distinguish the function from the usual ln, and because no base other than the natural base is used in the complex domain, making it the only Log function for complex numbers. The first 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 < \pi</math>, then

<math>\ log(z) = \ln(r) + i\phi \,\!</math>

where ln(r) is the usual natural logarithm of a real number. So defined, log is holomorphic for all complex numbers which are not non-positive reals, and it has the property

<math>\ e^{\log(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, log(e^z) does not always equal z, and log(zw) does not always equal log(z) + log(w).

A somewhat more natural definition of log(z) interprets it as a multi-valued function: for <math>\ z = r e^{i \phi}</math> we set

<math>\log(z) = \ln(r) + i(\phi + 2 \pi k) \,\!</math> : where k is 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 log 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 \,\left( \frac{1}{1} - x\,\left(\frac{1}{2} - x \,\left(\frac{1}{3} - x \,\left(\frac{1}{4} - x \,\left(\frac{1}{5}- \ldots \right)\right)\right)\right)\right) \quad{\rm for}\quad \left|x\right|<1.\,\!</math>

To obtain a better rate of convergence, the following identity can be used.

<math>\ln(x) = \ln\left(\frac{1+y}{1-y}\right) = 2\,y\, \left( \frac{1}{1} + \frac{1}{3} y^{2} + \frac{1}{5} y^{4} + \frac{1}{7} y^{6} + \frac{1}{9} y^{8} + \ldots \right) </math>

<math>\ln(x) = \ln\left(\frac{1+y}{1-y}\right) = 2\,y\, \left( \frac{1}{1} + y^{2} \, \left( \frac{1}{3} + y^{2} \, \left( \frac{1}{5} + y^{2} \, \left( \frac{1}{7} + y^{2} \, \left( \frac{1}{9} + \ldots \right) \right) \right)\right) \right) </math>

provided that y = (x−1)/(x+1) and x > 1.

For ln(x) where x > 1, the closer the value of x is to 1, the faster the rate of convergence. The identities associated with the logarithm can be leveraged to exploit this:

<math>\ln(123.456) = \ln(1.23456 \times 10^2) = \ln(1.23456) + \ln(10^2) \,\!</math>

<math>\ln(123.456) = \ln(1.23456) + 2 \times \ln(10) \,\!</math>

<math>\ln(123.456) \approx \ln(1.23456) + 2 \times 2.3025851 \,\!</math>

Such techniques were used before calculators, by referring to numerical tables and performing manipulations such as those above.

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>

In other words,

<math>\int { dx \over x} = \ln|x| + C</math>

and

<math>\int { {f^'(x) \over f(x)}\, dx} = \ln |f(x)| + C.</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

• John Napier
• Logarithmic integral function

References

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