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Encyclopedia > 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 (see the x-intercept of the graph). 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. For hyperbole, the figure of speech, see hyperbole. ... Logarithms to various bases: is to base e, is to base 10, and is to base 1. ... In mathematics, the base or radix is the number of various unique symbols (digits), including zero, that a positional numeral system uses to represent numbers in a given counting system. ... e is the unique number such that the value of the derivative of f (x) = ex (blue curve) at the point x = 0 is exactly 1. ... In mathematics, an irrational number is any real number that is not a rational number â€” that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... 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 = âˆ’1. ...

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 (made by me) File links The following pages link to this file: Natural logarithm Categories: GFDL images ... e is the unique number such that the value of the derivative of f (x) = ex (blue curve) at the point x = 0 is exactly 1. ... Natural logarithm Image File history File links Euler's_formula. ... Applications in Compound interest · Euler's identity & Euler's formula  · Half lives & Exponential growth/decay Compound interest refers to the fact that whenever interest is calculated, it is based not only on the original principal, but also on any unpaid interest that has been added to the principal. ... For other meanings, see List of topics named after Leonhard Euler In mathematical analysis, Eulers identity, named after Leonhard Euler, is the equation where is Eulers number, the base of the natural logarithm, is the imaginary unit, one of the two complex numbers whose square is negative one... Eulers 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. ... Half-Life For a quantity subject to exponential decay, the half-life is the time required for the quantity to fall 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 functions current size. ... A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value. ... Defining e Proof that e is irrational  · Representations of e · Lindemann–Weierstrass theorem In mathematics, the series expansion of the number e can be used to prove that e is irrational. ... The mathematical constant e can be represented in a variety of ways as a real number. ... In mathematics, the Lindemannâ€“Weierstrass theorem states that if Î±1,...,Î±n are algebraic numbers which are linearly independent over the rational numbers, then are algebraically independent over the algebraic numbers; in other words the set has transcendence degree n over . ... People John Napier  · Leonhard Euler For other people with the same name, see John Napier (disambiguation). ... Leonhard Paul Euler (pronounced Oiler; IPA ) (April 15, 1707 â€“ September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. ... Schanuel's conjecture Schanuels conjecture is that given any set of n complex numbers which have linear independence over the rational numbers, the set (up to twice the size) has transcendence degree of at least n over the rationals. ...

The natural logarithm function can also be defined as the inverse function of the exponential function, leading to the identities: In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ... The exponential function is one of the most important functions in mathematics. ...

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

In other words, the logarithm function is a bijection from the set of positive real numbers to the set of all real numbers. More precisely it is an isomorphism from the group of positive real numbers under multiplication to the group of real numbers under addition. A bijective function. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ... This picture illustrates how the hours on a clock form a group under modular 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.

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

In mathematics, the common logarithm is the logarithm with base 10. ... In mathematics, the common logarithm is the logarithm with base 10. ... Computer science, or computing science, is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. ... Plot of log2 x In mathematics, the binary logarithm (log2 n) is the logarithm for base 2. ... A programming language is an artificial language that can be used to control the behavior of a machine, particularly a computer. ... C is a general-purpose, block structured, procedural, imperative computer programming language developed in 1972 by Dennis Ritchie at the Bell Telephone Laboratories for use with the Unix operating system. ... C++ (pronounced see plus plus, IPA: ) is a general-purpose programming language with high-level and low-level capabilities. ... Fortran (previously FORTRAN[1]) is a general-purpose[2], procedural,[3] imperative programming language that is especially suited to numeric computation and scientific computing. ... BASIC (Beginners All-purpose Symbolic Instruction Code) is a family of high-level programming languages. ... A calculator is a device for performing calculations. ...

## 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 the 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, and second, because the natural logarithm can be defined quite easily using a simple integral or Taylor series — 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: Decimal, or denary, notation is the most common way of writing the base 10 numeral system, which uses various symbols for ten distinct quantities (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, called digits) together with the decimal point and the sign symbols + (plus) and &#8722; (minus... As the degree of the Taylor series rises, it approaches the correct function. ... For a non-technical overview of the subject, see Calculus. ...

$frac{d}{dx}log_b(x) =frac{log_b e}{x}$

If the base b is equal to e then the derivative is simply 1/x, and at x = 1 the slope of the graph is 1. In mathematics, the base or radix is the number of various unique symbols (digits), including zero, that a positional numeral system uses to represent numbers in a given counting system. ...

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. In fact, Nicholas Mercator first described them as log naturalis before calculus was even conceived. Nicholas (Nikolaus) Mercator (c. ...

## Definitions

Formally, ln(a) may be defined as the area under the graph (integral) of 1/x from 1 to a, that is, The integral of f(x) from a to b is the area above the x-axis and below the curve y = f(x), minus the area below the x-axis and above the curve, for x in the interval [a,b]. Integration is a core concept of advanced mathematics, specifically...

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

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. e is the unique number such that the value of the derivative of f (x) = ex (blue curve) at the point x = 0 is exactly 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 $e^{ln(x)} = x!$. 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. The exponential function is one of the most important functions in mathematics. ... In mathematics, a series is a sum of a sequence of terms. ... In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ...

## Derivative, Taylor series

The derivative of the natural logarithm is given by For a non-technical overview of the subject, see Calculus. ...

$frac{d}{dx} ln(x) = frac{1}{x}.,$

This leads to the Taylor series As the degree of the Taylor series rises, it approaches the correct function. ...

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

which is also known as the Mercator series. In mathematics, the Mercator series or Newton-Mercator series is the Taylor series for the natural logarithm. ...

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

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

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: In mathematics, in the area of combinatorics, the binomial transform is a transformation of sequence by computing its forward differences. ...

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

This series is similar to a BBP-type formula. In mathematics, the Bailey-Borwein-Plouffe formula (BBP formula) is a Ï€ summation formula discovered in 1995 by Simon Plouffe. ...

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)|). This is the case because of the chain rule and the following fact: The integral of f(x) from a to b is the area above the x-axis and below the curve y = f(x), minus the area below the x-axis and above the curve, for x in the interval [a,b]. Integration is a core concept of advanced mathematics, specifically... In calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose derivative is equal to f, i. ... In calculus, the chain rule is a formula for the derivative of the composite of two functions. ...

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

In other words,

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

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 indefinite integral of a given function (i. ...

The natural logarithm can be integrated using integration by parts: In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, possibly simpler, integrals. ...

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

## Numerical value

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

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

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

 $ln(x) = lnleft(frac{1+y}{1-y}right)$ $= 2,y, left( frac{1}{1} + frac{1}{3} y^{2} + frac{1}{5} y^{4} + frac{1}{7} y^{6} + frac{1}{9} y^{8} + ldots right)$ $= 2,y, left( frac{1}{1} + y^{2} , left( frac{1}{3} + y^{2} , left( frac{1}{5} + y^{2} , left( frac{1}{7} + y^{2} , left( frac{1}{9} + ldots right) right) right)right) right)$
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, Newtons method (also known as the Newtonâ€“Raphson method or the Newtonâ€“Fourier method) is an efficient algorithm for finding approximations to the zeros (or roots) of a real-valued function. ...

An alternative for extremely high precision calculation is the formula [citation needed]

$ln x approx frac{pi}{2 Mleft(1, frac{4}{s}right)} - m ln 2$

where M denotes the arithmetic-geometric mean and In mathematics, the arithmetic-geometric mean M(x, y) of two positive real numbers x and y is defined as follows: we first form the arithmetic mean of x and y and call it a1, i. ...

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

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.) When a circles diameter is 1, its circumference is Ï€. Pi or Ï€ is the ratio of a circles circumference to its diameter in Euclidean geometry, approximately 3. ...

### 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. The following tables list the computational complexity of various algorithms for common mathematical operations. ... Complexity theory is part of the theory of computation dealing with the resources required during computation to solve a given problem. ...

## 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. The natural logarithm is the logarithm to the base e, where e is equal to 2. ... 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 = âˆ’1. ...

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. In mathematics, the complex plane is a way of visualising the space of the complex numbers. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ...

For other people with the same name, see John Napier (disambiguation). ... In mathematics, the logarithmic integral function or integral logarithm li(x) is a special function. ... Nicholas (Nikolaus) Mercator (c. ... The polylogarithm (also known as de JonquiÃ¨res function) is a special function Lis(z) that is defined by the sum The above definition is valid for all complex numbers s and z where |z|< 1. ... The von Mangoldt function is an arithmetic function named after German mathematician Hans von Mangoldt. ... e is the unique number such that the value of the derivative of f (x) = ex (blue curve) at the point x = 0 is exactly 1. ...

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