# natural logarithm

The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is an irrational constant approximately equal to 2.718281828459. In simple terms, the natural logarithm of a number x is the power to which e would have to be raised to equal x — for example the natural log of e itself is 1 because e1 = e, while the natural logarithm of 1 would be 0, since e0 = 1. The natural logarithm can be defined for all positive real numbers x as the area under the curve y = 1/t from 1 to x, and can also be defined for non-zero complex numbers as explained below.
Graph of the natural logarithm function. The function goes to negative infinity as x approaches 0, but grows slowly to positive infinity as x increases in value.

 Part of a series of articles on The mathematical constant, e Natural logarithm Applications in Compound interest Euler's identity & Euler's formula Half lives & Exponential growth/decay Defining e Proof that e is irrational Representations of e Lindemann–Weierstrass theorem People John Napier Leonhard Euler Schanuel's conjecture

The natural logarithm function can also be defined as the inverse function of the exponential function, leading to the identities:

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 are useful for solving equations in which the unknown appears as the exponent of some other quantity.

## Notational conventions

• Mathematicians, statisticians, and some engineers generally understand either "log(x)" or "ln(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.
• Some engineers, biologists, and some others generally write "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 case of some computer scientists, log2(x).
• In most commonly-used programming languages, including C, C++, Fortran, and BASIC, "log" or "LOG" refers to the natural logarithm.
• In hand-held calculators, the natural logarithm is denoted ln, whereas log is the base-10 logarithm.

## Why it is called "natural"

Initially, it might seem that since we use base 10 for nearly all calculations, this base would be more "natural" than base e, but there are several senses in which loge is more "natural". First, throughout the natural and mathematical sciences variables appear as the exponents of e in many more important expressions than as exponents of 10—the only thing special about 10, after all, is the accident that it happens to be the number of fingers with which most humans are born. Thus, the natural logarithm is almost always more useful in practice. As a related example, consider the problem of differentiating a logarithmic function:
If the base b equals e, then the derivative is simply 1/x, and at x = 1 this derivative equals 1. Another sense in which the base-e logarithm is the most natural is that it can be defined quite easily in terms of a simple integral or Taylor series and this is not true of other logarithms.

Further senses of this naturalness make no use of calculus. As an example, there are a number of simple series involving the natural logarithm. In fact, Pietro Mengoli and Nicholas Mercator called it logarithmus naturalis a few decades before Newton and Leibniz developed calculus.[1]

## Definitions

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

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

This can be demonstrated by letting as follows:

The number e can then be defined as the unique real number a such that ln(a) = 1.

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

## Derivative, Taylor series

The derivative of the natural logarithm is given by
This leads to the Taylor series

which is also known as the Mercator series.

Substituting x-1 for x, we obtain an alternative form for ln(x) itself, namely
[2]

By using the Euler transform on the Mercator series, one obtains the following, which is valid for any x with absolute value greater than 1:

This series is similar to a BBP-type formula.

Also note that is its own inverse function, so to yield the natural logarithm of a certain number n, simply put in for x.

## 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:

In other words,

and

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

Letting f(x) = cos(x) and f'(x)= - sin(x):

where C is an arbitrary constant of integration.

The natural logarithm can be integrated using integration by parts:

## Numerical value

To calculate the numerical value of the natural logarithm of a number, the Taylor series expansion can be rewritten as:

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

provided that y = (x−1)/(x+1) and x > 0.

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:

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

### High precision

To compute the natural logarithm with many digits of precision, the Taylor series approach is not efficient since the convergence is slow. An alternative is to use Newton's method to invert the exponential function, whose series converges more quickly.

An alternative for extremely high precision calculation is the formula

where M denotes the arithmetic-geometric mean and

with m chosen so that p bits of precision is attained. In fact, if this method is used, Newton inversion of the natural logarithm may conversely be used to calculate the exponential function efficiently. (The constants ln 2 and π can be pre-computed to the desired precision using any of several known quickly converging series.)

### Computational complexity

See main article: Computational complexity of mathematical operations
The computational complexity of computing the natural logarithm (using the arithmetic-geometric mean) is O(M(n) ln n). Here n is the number of digits of precision at which the natural logarithm is to be evaluated and M(n) is the computational complexity of multiplying two n-digit numbers.

## Complex logarithms

Main article: Complex logarithm
The exponential function can be extended to a function which gives a complex number as ex for any arbitrary complex number x; simply use the infinite series with x complex. This exponential function can be inverted to form a complex logarithm that exhibits most of the properties of the ordinary logarithm. There are two difficulties involved: no x has ex = 0; and it turns out that e2πi = 1 = e0. Since the multiplicative property still works for the complex exponential function, ez = ez+2nπi, for all complex z and integers n.

So the logarithm cannot be defined for the whole complex plane, and even then it is multi-valued – any complex logarithm can be changed into an "equivalent" logarithm by adding any integer multiple of 2πi at will. The complex logarithm can only be single-valued on the cut plane. For example, ln i = 1/2 πi or 5/2 πi or −3/2 πi, etc.; and although i4 = 1, 4 log i can be defined as 2πi, or 10πi or −6 πi, and so on.

<gallery caption="Plots of the natural logarithm function on the complex plane (principal branch)"> Image:NaturalLogarithmRe.png| z = Re(ln(x+iy)) Image:NaturalLogarithmIm.png| z = |Im(ln(x+iy))| Image:NaturalLogarithmAbs.png| z = |ln(x+iy)| Image:NaturalLogarithmAll.png| Superposition of the previous 3 graphs </gallery>

## References

hyperbola (Greek ὑπερβολή literally 'overshooting' or 'excess') is a type of conic section defined as the intersection between a right circular conical surface and a plane which cuts through both halves
logarithm (to base b) of a number x is the exponent y that satisfies x = by. It is written logb(x) or, if the base is implicit, as log(x).
base refers to the number b in an expression of the form bn. The number n is called the exponent and the expression is known formally as exponentiation of b by n or the exponential of n with base
e is the unique real number such that the value of the derivative (slope of the tangent line) of f(x) = ex at the point x = 0 is exactly 1.
In mathematics, an irrational number is any real number that is not a rational number — that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero.
In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2.4871773339…. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as π and
INTErnational Gamma-Ray Astrophysics Laboratory (INTEGRAL) is detecting some of the most energetic radiation that comes from space. It is the most sensitive gamma ray observatory ever launched.
In mathematics, a complex number is a number of the form

where a and b are real numbers, and i is the imaginary unit, with the property i ² = −1.
e is the unique real number such that the value of the derivative (slope of the tangent line) of f(x) = ex at the point x = 0 is exactly 1.
Compound interest is the concept of adding accumulated interest back to the principal, so that interest is earned on interest from that moment on. The act of declaring interest to be principal is called compounding (i.e. interest is compounded).
Euler's identity, named after Leonhard Euler, is the equation

where
is Euler's number, the base of the natural logarithm,

Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. (Euler's identity is a special case of the Euler formula.
For , see .
The half-life of a quantity, subject to exponential decay, is the time required for the quantity to decay to half of its initial value.
In mathematics, exponential growth (or geometric growth) occurs when the growth rate of a function is always proportional to the function's current size. Such growth is said to follow an exponential law (but see also Malthusian growth model).
A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value. Symbolically, this can be expressed as the following differential equation, where N is the quantity and λ is a positive number called the decay constant.
In mathematics, the series representation of Euler's number e

can be used to prove that e is irrational. Of the many representations of e, this is the Taylor series for the exponential function ey
The mathematical constant e can be represented in a variety of ways as a real number. Since e is an irrational number (see proof that e is irrational), it cannot be represented as a fraction, but it can be represented as a continued fraction.
John Napier

Painting of John Napier
Born 1550
Merchiston Tower, in
Edinburgh, Scotland
Died 4 April 1617
Leonhard Euler

Portrait by Johann Georg Brucker
Born March 15 1707
Basel, Switzerland
Died September 18 [O.S.
In mathematics, specifically transcendence theory, Schanuel's conjecture is the following statement:
Given any n complex numbers z1,...

inverse function for ƒ, denoted by ƒ−1, is a function in the opposite direction, from B to A, with the property that a round trip (a composition) returns each element to itself.
The exponential function is one of the most important functions in mathematics. The application of this function to a value x is written as exp(x).
In mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property that, for every y in Y, there is exactly one x in X such that
f(x) = y.
In mathematics, an isomorphism (Greek: isos "equal", and morphe "shape") is a bijective map f such that both f and its inverse f −1 are homomorphisms, i.e., structure-preserving mappings.
group is a set with a binary operation that satisfies certain axioms, detailed below. For example, the set of integers with addition is a group. The branch of mathematics which studies groups is called group theory.
In mathematics, the common logarithm is the logarithm with base 10. It is also known as the decadic logarithm, named after its base. It is indicated by log10(x), or sometimes Log(x) with a capital L