Natural logarithm

From Wikipedia, the free encyclopedia

"Base e" redirects here. For the numbering system which uses e as its base, see Non-integer representation#Base e.

Graph of the natural logarithm function. The function slowly grows to positive infinity as x increases and rapidly goes to negative infinity as x approaches 0 ("slowly" and "rapidly" as compared to any power law of x); the y-axis is an asymptote.

The natural logarithm is the logarithm to the base e, where e is an irrationaland transcendental constant approximately equal to 2.718281828. The natural logarithm is generally written as ln(x), loge(x) or sometimes, if the base of e is implicit, as simply log(x).[1] The natural logarithm of a number x is the power to which e would have to be raised to equal x. For example, ln(7.389...) is 2, because e2=7.389.... The natural log of e itself (ln(e)) is 1 because e1 = e, while the natural logarithm of 1 (ln(1)) is 0, since e0 = 1. The natural logarithm can be defined for any positive real number a as the area under the curve y = 1/x from 1 to a. The simplicity of this definition, which is matched in many other formulas involving the natural logarithm, leads to the term "natural." The definition can be extended to non-zero complex numbers, as explained below. The natural logarithm function, if considered as a real-valued function of a real variable, is the inverse function of the exponential function, leading to the identities:

Like all logarithms, the natural logarithm maps multiplication into addition:

Thus, the logarithm function is an isomorphism from the group of positive real numbers under multiplication to the group of real numbers under addition, represented as a function:

Logarithms can be defined to any positive base other than 1, not just e; however logarithms in other bases differ only by a constant multiplier from the natural logarithm, and are usually defined in terms of the latter. Logarithms are useful for solving equations in which the unknown appears as the exponent of some other quantity. For example, logarithms are used to solve for the half-life, decay constant, or unknown time in exponential decay problems. They are important in many branches of mathematics and the sciences and are used in finance to solve problems involving compound interest.
Part of a series of articles on

The mathematical constant e

Natural logarithm Exponential function

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

Contents
[hide]

1 History 2 Notational conventions 3 Origin of the term natural logarithm 4 Definitions 5 Properties 6 Derivative, Taylor series 7 The natural logarithm in integration 8 Numerical value

o o o

8.1 Natural logarithm of 10 8.2 High precision 8.3 Computational complexity

9 Continued fractions 10 Complex logarithms 11 See also 12 References 13 External links

[edit]History
The first mention of the natural logarithm was by Nicholas Mercator in his workLogarithmotechnia published in 1668,[2] although the mathematics teacher John Speidell had already in 1619 compiled a table on the natural logarithm.[3] It was formerly also called hyperbolic logarithm,[4] as it corresponds to the area under a hyperbola. It is also sometimes referred to as the Napierian logarithm, although the original meaning of this term is slightly different.

[edit]Notational

conventions

The notations ln x and loge x both refer unambiguously to the natural logarithm of x. Log x without an explicit base may also refer to the natural logarithm. This usage is common in some scientific contexts as well as in many programming languages.[5] Log x is frequently used to denote the common (base 10) logarithm, however.

[edit]Origin

of the term natural logarithm

Initially, it might seem that since the common numbering system is base 10, this base would be more "natural" than base e. But mathematically, the number 10 is not particularly significant. Its use culturallyas the basis for many societies numbering systemslikely arises from humans typical number of fingers.[6] Other cultures have based their counting systems on such choices as 5, 8, 12,20, and 60.[7][8][9] 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:[10]

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. Pietro Mengoli and Nicholas Mercator called it logarithmus naturalis a few decades before Newton and Leibniz developed calculus.[11]

[edit]Definitions

ln(a) illustrated as the area under the curve f(x) = 1/x from 1 to a. If a is less than 1, the area from a to 1 is counted as negative.

Formally, ln(a) may be defined as the integral,

This function is a logarithm because it satisfies the fundamental property of a logarithm:

This can be demonstrated by splitting the integral that defines ln(ab) into two parts and then making the variable substitution x = ta in the second part, 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, say by using an infinite series, the natural logarithm may be defined as its inverse function, i.e., ln 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.

[edit]Properties

(see complex logarithm)

The number e is an important mathematical constant, approximately equal to 2.71828, that is the base of [1] n the natural logarithm. It is the limit of (1 + 1/n) as n becomes large, an expression that arises in the [2] study of compound interest, and can also be calculated as the sum of the infinite series

The constant can be defined in many ways; for example, e is the unique real numbersuch that the x value of the derivative (slope of the tangent line) of the function f(x) = e at the point x = 0 is equal to [3] x 1. The function e so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base e. The natural logarithm of a positive number k can also be defined directly as the area under the curve y = 1/x between x = 1 and x = k, in which case, e is the number whose natural logarithm is 1. There are also more alternative characterizations. Sometimes called Euler's number after the Swiss mathematician Leonhard Euler, eis not to be confused with the EulerMascheroni constant, sometimes called simply Euler's constant. The number e is also known as Napier's constant, but Euler's choice of this symbol is said to have been [4] retained in his honor. The numbere is of eminent importance in [5] mathematics, alongside 0, 1, and i. All five of these numbers play important and recurring roles across mathematics, and are the five constants appearing in one formulation of Euler's identity. Like the constant , e isirrational: it is not a ratio of integers; and it is transcendental: it is not a root of anynon-zero polynomial with rational coefficients. The numerical value of e truncated to 50 decimal places is 2.71828182845904523536028747135266249775724709369995... (sequence A001113 in OEIS).
Part of a series of articles on

The mathematical constant e

Natural logarithm Exponential function

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

Contents
[hide]

1 History 2 Applications

o o o o

2.1 Compound interest 2.2 Bernoulli trials 2.3 Derangements 2.4 Asymptotics

3 e in calculus

3.1 Alternative characterizations

4 Properties

o o o o o

4.1 Calculus 4.2 Exponential-like functions 4.3 Number theory 4.4 Complex numbers 4.5 Differential equations

5 Representations

o o

5.1 Stochastic representations 5.2 Known digits

6 In computer culture 7 Notes 8 Further reading 9 External links

[edit]History The first references to the constant were published in 1618 in the table of an appendix of a work [6] on logarithms by John Napier. However, this did not contain the constant itself, but simply a list of logarithms calculated from the constant. It is assumed that the table was written by William

Oughtred. The discovery of the constant itself is credited to Jacob Bernoulli, who attempted to find the value of the following expression (which is in fact e):

The first known use of the constant, represented by the letter b, was in correspondence from Gottfried Leibniz to Christiaan Huygens in 1690 and 1691. Leonhard Euler introduced the letter e as the base for natural logarithms, writing in a letter to Christian Goldbach of 25 [7] November 1731. Euler started to use the letter e for the constant in 1727 or 1728, in an [8] unpublished paper on explosive forces in cannons, and the first appearance of e in a publication was Euler's Mechanica (1736). While in the subsequent years some researchers used the letter c, e was more common and eventually became the standard. [edit]Applications [edit]Compound

interest

The effect of earning 20% annual interest on an initial $1,000 investment at various compounding frequencies

Jacob Bernoulli discovered this constant by studying a question aboutcompound interest: An account starts with $1.00 and pays 100 percent interest per year. If the interest is credited

[6]

once, at the end of the year, the value of the account at year-end will be $2.00. What happens if the interest is computed and credited more frequently during the year? If the interest is credited twice in the year, the interest rate for each 6 months will be 2 50%, so the initial $1 is multiplied by 1.5 twice, yielding $1.001.5 = $2.25 at the end of 4 the year. Compounding quarterly yields $1.001.25 = $2.4414..., and compounding 12 monthly yields $1.00(1+1/12) = $2.613035... If there are n compounding intervals, the interest for each interval will be 100%/n and the value at the end of the year will be n $1.00(1 + 1/n) .

Bernoulli noticed that this sequence approaches a limit (the force of interest) with larger n and, thus, smaller compounding intervals. Compounding weekly (n = 52) yields $2.692597..., while compounding daily (n = 365) yields $2.714567..., just two cents more. The limit as n grows large is the number that came to be known as e; withcontinuous compounding, the account value will reach $2.7182818.... More generally, an account that starts at $1 and offers an annual interest rate of R will, Rt after t years, yield e dollars with continuous compounding. (Here R is a fraction, so for 5% interest,R = 5/100 = 0.05) [edit]Bernoulli

trials

The number e itself also has applications to probability theory, where it arises in a way not obviously related to exponential growth. Suppose that a gambler plays a slot machine that pays out with a probability of one in n and plays it n times. Then, for large n (such as a million) the probability that the gambler will lose every bet is (approximately) 1/e. For n = 20 it is already 1/2.72. This is an example of a Bernoulli trials process. Each time the gambler plays the slots, there is a one in one million chance of winning. Playing one million times is modelled by the binomial distribution, which is closely related to the binomial theorem. The probability of winning k times out of a million trials is;

In particular, the probability of winning zero times (k = 0) is

This is very close to the following limit for 1/e:

[edit]Derangements Another application of e, also discovered in part by Jacob Bernoulli along with Pierre Raymond de Montmort is in the problem ofderangements, also [9] known as the hat check problem: n guests are invited to a party, and at the door each guest checks his hat with the butler who then places them into n boxes, each labelled with the name of one guest. But the butler does not know the identities of the guests, and so he puts the hats into boxes selected at random. The problem of de Montmort is to find the probability that none of the hats gets put into the right box. The answer is:

As the number n of guests tends to infinity, pn approaches 1/e. Furthermore, the number of ways the hats can be placed into the boxes so that none of the hats is in the right box is n!/e rounded to the [10] nearest integer, for every positive n. [edit]Asymptotics The number e occurs naturally in connection with many problems involving asymptotics. A prominent example is Stirling's formula for the asymptotics of the factorial function, in which both the numbers e and enter:

A particular consequence of this is

. [edit]

e in calculus

The natural log at e, ln(e), is equal to 1

The principal motivation for introducing the number e, particularly in calculus, is to performdifferential and integral [11] calculus with exponential functions and logarithms. A x general exponential function y = a has derivative given as the limit:

The limit on the right-hand side is independent of the variable x: it depends only on the basea. When the base is e, this limit is equal to one, and so e is symbolically defined by the equation:

Consequently, the exponential function with base e is particularly suited to doing calculus. Choosing e, as opposed to some other number, as the base of the exponential function makes calculations involving the derivative much simpler. Another motivation comes from considering the basea logarithm.[12] Considering the definition of the derivative of loga x as the limit:

where the substitution u = h/x was made in the last step. The last limit appearing in this calculation is again an undetermined limit that depends only on the base a, and if that base is e, the limit is one. So symbolically,

The logarithm in this special base is called the natural logarithm and is represented as ln; it behaves well under differentiation since there is no undetermined limit to carry through the calculations. There are thus two ways in which to select a special number a = e. One way is to set the derivative of the exponential x x function a to a , and solve for a. The other way is to set the derivative of the base a logarithm to 1/x and solve for a. In each case, one arrives at a convenient choice of base for doing calculus. In fact, these two solutions for a are actually the same, the number e.

[edit]Alternative

characterizations

The area between the x-axis and the graph y = 1/x, between x = 1 and x = e is 1.

See also: Representations of e Other characterizations of e are also possible: one is as the limit of a sequence, another is as the sum of an infinite series, and still others rely on integral calculus. So far, the following two (equivalent) properties have been introduced: 1. The number e is the unique positive real number such that

2. The number e is the unique positive real number such that

The following three characterizations can be proven equivalent: 3. The number e is the limit

Similarly:

4. The number e is the sum of the infinite series

where n! is the factorial of n. 5. The number e is the unique positive real number such that