The natural logarithm, formerly known as the hyperbolic logarithm[1], is the logarithm to the base e, where e is an irrational constant approximately equal to 2. In Geometry, a hyperbola ( Greek, "over-thrown" has several equivalent definitions In Mathematics, the logarithm of a number to a given base is the power or Exponent to which the base must be raised in order to produce radix|basis (topologyIn Arithmetic, the base refers to the number b in an expression of the form b n. The Mathematical constant e is the unique Real number such that the function e x has the same value as the slope of the tangent line 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 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. In Mathematics, the real numbers may be described informally in several different ways The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted

Graph of the natural logarithm function. The function quickly goes to negative infinity as x approaches 0, but grows slowly to positive infinity as x increases.

Part of a series of articles onThe mathematical constant,

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:

$e^{\ln(x)} = x \qquad \mbox{if }x > 0\,\!$
$\ln(e^x) = x.\,\!$

In other words, the logarithm function is a bijection from the set of positive real numbers to the set of all real numbers. The Mathematical constant e is the unique Real number such that the function e x has the same value as the slope of the tangent line Compound interest is the concept of adding accumulated Interest back to the principal so that interest is earned on interest from that moment on In Mathematical analysis, Euler's identity, named after Leonhard Euler, is the equation e^{i \pi} + 1 = 0 \\! where This article is about Euler's formula in Complex analysis. For Euler's formula in algebraic topology and polyhedral combinatorics see Euler characteristic Half-Life (computer-game page here It's already listed in the disambiguation page Exponential growth (including Exponential decay) occurs when the growth rate of a mathematical function is proportional to the function's current value A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value In Mathematics, the series representation of Euler's number e e = \sum_{n = 0}^{\infty} \frac{1}{n!}\! can be used to prove The Mathematical constant ''e'' can be represented in a variety of ways as a Real number. In Mathematics, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers For other people with the same name see John Napier (disambiguation. In Mathematics, specifically Transcendence theory, Schanuel's conjecture is the following statement Given any n Complex numbers In Mathematics, if &fnof is a function from A to B then an inverse function for &fnof is a function in the opposite direction from B The exponential function is a function 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 More precisely it is an isomorphism from the group of positive real numbers under multiplication to the group of real numbers under addition. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element Represented as a function:

$\ln : \mathbb{R}^+ \to \mathbb{R}$

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. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function

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. The common logarithm is the Logarithm with base 10 It is also known as the decadic logarithm, named after its base

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) (although this is often written lg(x) instead). The common logarithm is the Logarithm with base 10 It is also known as the decadic logarithm, named after its base Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their In Mathematics, the binary logarithm (log2 n) is the Logarithm for Base 2

In most commonly-used programming languages, including C, C++, MATLAB, Fortran, and BASIC, "log" or "LOG" refers to the natural logarithm. A programming language is an Artificial language that can be used to write programs which control the behavior of a machine particularly a Computer. tags please moot on the talk page first! --> In Computing, C is a general-purpose cross-platform block structured C++ (" C Plus Plus " ˌsiːˌplʌsˈplʌs is a general-purpose Programming language. MATLAB is a numerical computing environment and Programming language. Fortran (previously FORTRAN) is a general-purpose, procedural, imperative Programming language that is especially suited to In Computer programming, BASIC (an Acronym for Beginner's All-purpose Symbolic Instruction Code) is a family of High-level programming languages

In hand-held calculators, the natural logarithm is denoted ln, whereas log is the base-10 logarithm. A calculator is device for performing mathematical calculations distinguished from a Computer by having a limited problem solving ability and an interface optimized for interactive

Why it is called “natural”

Initially, it might seem that since our numbering system is base 10, this base would be more “natural” than base e. The decimal ( base ten or occasionally denary) Numeral system has ten as its base. But mathematically, the number 10 is not particularly significant. Its use culturally — as the basis for many societies’ numbering systems — likely arises from humans’ typical number of fingers. [2] Other cultures have based their counting systems on such choices as 5, 20, and 60. [3][4][5]

Loge is a “natural” log because it automatically springs from, and appears so often, in mathematics. For example, consider the problem of differentiating a logarithmic function:

$\frac{d}{dx}\log_b(x) = \frac{\log_b(e)}{x} =\frac{1}{\ln(b)x}$

If the base b equals e, then the derivative is simply 1/x, and at x = 1 this derivative equals 1. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change radix|basis (topologyIn Arithmetic, the base refers to the number b in an expression of the form b n. 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. In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives

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. Pietro Mengoli (1626-1686 was an Italian Mathematician and clergyman from Bologna, where he studied with Bonaventura Cavalieri at the University of Nicholas ( Nikolaus) Mercator (c 1620 Eutin -1687 Versailles) also known by his Germanic name Kauffmann, was a 17th-century mathematician Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements [6]

Definitions

Ln(x) defined as the area under the curve f(x) = 1/x.

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

$\ln(a)=\int_1^a \frac{1}{x}\,dx.$

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

$\ln(ab)=\ln(a)+\ln(b) \,\!$

This can be demonstrated by letting $t=\tfrac xa$ as follows:

$\ln (ab) = \int_1^{ab} \frac{1}{x} \; dx = \int_1^a \frac{1}{x} \; dx \; + \int_a^{ab} \frac{1}{x} \; dx =\int_1^{a} \frac{1}{x} \; dx \; + \int_1^{b} \frac{1}{t} \; dt = \ln (a) + \ln (b)$

The number e can then be defined as the unique real number a such that ln(a) = 1. The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space The Mathematical constant e is the unique Real number such that the function e x has the same value as the slope of the tangent line

Alternatively, if the exponential function has been defined first using an infinite series, the natural logarithm may be defined as its inverse function, i. The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with In Mathematics, if &fnof is a function from A to B then an inverse function for &fnof is a function in the opposite direction from B e. , ln(x) is that function such that $e^{\ln(x)} = x\!$. 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

$\frac{d}{dx} \ln(x) = \frac{1}{x}.\,$
The Taylor polynomials for loge(1+x) only provide accurate approximations in the range -1 < x ≤ 1. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change Note that, for x > 1, the Taylor polynomials of higher degree are worse approximations.

This leads to the Taylor series for ln(1 + x) around 0; also known as the Mercator series

$\ln(1+x)=\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} x^n = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots \quad{\rm for}\quad \left|x\right| \leq 1\quad$
${\rm unless}\quad x = -1$

At right is a picture of ln(1 + x) and some of its Taylor polynomials around 0. In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives In Mathematics, the Mercator series or Newton-Mercator series is the Taylor series for the Natural logarithm. In Calculus, Taylor's theorem gives a sequence of approximations of a Differentiable function around a given point by Polynomials (the Taylor These approximations converge to the function only in the region -1 < x ≤ 1; outside of this region the higher-degree Taylor polynomials are worse approximations for the function.

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

$\ln(x)=\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} (x-1) ^ n$
$\ln(x)= (x - 1) - \frac{(x-1) ^ 2}{2} + \frac{(x-1)^3}{3} - \frac{(x-1)^4}{4} \cdots$
${\rm for}\quad \left|x-1\right| \leq 1\quad {\rm unless}\quad x = 0.$[7]

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:

$\ln{x \over {x-1}} = \sum_{n=1}^\infty {1 \over {n x^n}} = {1 \over x}+ {1 \over {2x^2}} + {1 \over {3x^3}} + \cdots$

This series is similar to a BBP-type formula. In Combinatorial Mathematics the binomial transform is a Sequence transformation (ie a transform of a Sequence) that computes its Forward

Also note that $x \over {x-1}$ is its own inverse function, so to yield the natural logarithm of a certain number n, simply put in $n \over {n-1}$ 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)|). The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space In Calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose Derivative This is the case because of the chain rule and the following fact:

$\ {d \over dx}\left( \ln \left| x \right| \right) = {1 \over x}.$

In other words,

$\int { 1 \over x} dx = \ln|x| + C$

and

$\int { \frac{f'(x)}{f(x)}\, dx} = \ln |f(x)| + C.$

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

$\int \tan (x) \,dx = \int {\sin (x) \over \cos (x)} \,dx$
$\int \tan (x) \,dx = \int {-{d \over dx} \cos (x) \over {\cos (x)}} \,dx.$

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

$\int \tan (x) \,dx = -\ln{\left| \cos (x) \right|} + C$
$\int \tan (x) \,dx = \ln{\left| \sec (x) \right|} + C$

where C is an arbitrary constant of integration. In Calculus, the chain rule is a Formula for the Derivative of the composite of two functions. In Calculus, the Indefinite integral of a given function (ie the set of all Antiderivatives of the function is always written with a constant the constant

The natural logarithm can be integrated using integration by parts:

$\int \ln (x) \,dx = x \ln (x) - x + C.$

Numerical value

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

$\ln(1+x)= x \,\left( \frac{1}{1} - x\,\left(\frac{1}{2} - x \,\left(\frac{1}{3} - x \,\left(\frac{1}{4} - x \,\left(\frac{1}{5}- \ldots \right)\right)\right)\right)\right) \quad{\rm for}\quad \left|x\right|<1.\,\!$

To obtain a better rate of convergence, the following identity can be used. In Calculus, and more generally in Mathematical analysis, integration by parts is a rule that transforms the Integral of products of functions into other

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

$\ln(123.456)\!$  $= \ln(1.23456 \times 10^2) \,\!$
$= \ln(1.23456) + \ln(10^2) \,\!$
$= \ln(1.23456) + 2 \times \ln(10) \,\!$
$\approx \ln(1.23456) + 2 \times 2.3025851 \,\!$

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. In Numerical analysis, Newton's method (also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson) is perhaps the

An alternative for extremely high precision calculation is the formula

$\ln x \approx \frac{\pi}{2 M(1,4/s)} - m \ln 2$

where M denotes the arithmetic-geometric mean and

$s = x \,2^m > 2^{p/2},$

with m chosen so that p bits of precision is attained. In Mathematics, the arithmetic-geometric mean (AGM of two positive Real numbers x and y is defined as follows First compute the Arithmetic 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. IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems )

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). The following tables list the Computational complexity of various Algorithms for common Mathematical operations Here complexity refers to the Time complexity Computational complexity theory, as a branch of the Theory of computation in Computer science, investigates the problems related to the amounts of resources 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. In Complex analysis, the complex logarithm is the extension of the Natural logarithm function ln( x) &ndash originally defined for Real numbers Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted 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. In Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis The complex logarithm can only be single-valued on the cut plane. In Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis 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.

z = Re(ln(x+iy))  z = |Im(ln(x+iy))|  z = |ln(x+iy)|  Superposition of the previous 3 graphs
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