History of logarithms

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https://en.wikipedia.org/wiki/History_of_logarithms 

 

Title page of John Napier's Mirifici Logarithmorum Canonis Descriptio from 1614, the first published table of logarithms

 

A page from Napier's Mirifici logarithmorum tables, with trigonometric and log trig data for 34 degrees

The history of logarithms is the story of a correspondence (in modern terms, a group isomorphism) between multiplication on the positive real numbers and addition on the real number line that was formalized in seventeenth century Europe and was widely used to simplify calculation until the advent of the digital computer. The Napierian logarithms were published first in 1614. E. W. Hobson called it "one of the very greatest scientific discoveries that the world has seen." Henry Briggs introduced common (base 10) logarithms, which were easier to use. Tables of logarithms were published in many forms over four centuries. The idea of logarithms was also used to construct the slide rule, which became ubiquitous in science and engineering until the 1970s. A breakthrough generating the natural logarithm was the result of a search for an expression of area against a rectangular hyperbola, and required the assimilation of a new function into standard mathematics.

Napier's wonderful invention

The method of logarithms was publicly propounded for the first time by John Napier in 1614, in his book entitled Mirifici Logarithmorum Canonis Descriptio (Description of the Wonderful Canon of Logarithms). The book contains fifty-seven pages of explanatory matter and ninety pages of tables of trigonometric functions and their natural logarithms. These tables greatly simplified calculations in spherical trigonometry, which are central to astronomy and celestial navigation and which typically include products of sines, cosines and other functions. Napier described other uses, such as solving ratio problems, as well.

John Napier wrote a separate volume describing how he constructed his tables, but held off publication to see how his first book would be received. John died in 1617. His son, Robert, published his father's book, Mirifici Logarithmorum Canonis Constructio (Construction of the Wonderful Canon of Logarithms), with additions by Henry Briggs, in 1620.

Napier conceived the logarithm as the relationship between two particles moving along a line, one at constant speed and the other at a speed proportional to its distance from a fixed endpoint. While in modern terms, the logarithm function can be explained simply as the inverse of the exponential function or as the integral of 1/x, Napier worked decades before calculus was invented, the exponential function was understood, or coordinate geometry was developed by Descartes. Napier pioneered the use of a decimal point in numerical calculation, something that did not become commonplace until the next century.

Napier's new method for computation gained rapid acceptance. Johannes Kepler praised it; Edward Wright, an authority on navigation, translated Napier's Descriptio into English the next year. Briggs extended the concept to the more convenient base 10.

Common logarithm

Main article: Common logarithm

 

Canon logarithmorum

As the common log of ten is one, of a hundred is two, and a thousand is three, the concept of common logarithms is very close to the decimal-positional number system. The common log is said to have base 10, but base 10,000 is ancient and still common in East Asia. In his book The Sand Reckoner, Archimedes used the myriad as the base of a number system designed to count the grains of sand in the universe. As was noted in 2000:

In antiquity Archimedes gave a recipe for reducing multiplication to addition by making use of geometric progression of numbers and relating them to an arithmetic progression.

In 1616 Henry Briggs visited John Napier at Edinburgh in order to discuss the suggested change to Napier's logarithms. The following year he again visited for a similar purpose. During these conferences the alteration proposed by Briggs was agreed upon, and on his return from his second visit to Edinburgh, in 1617, he published the first chiliad of his logarithms.

In 1624, Briggs published his Arithmetica Logarithmica, in folio, a work containing the logarithms of thirty thousand natural numbers to fourteen decimal places (1-20,000 and 90,001 to 100,000). This table was later extended by Adriaan Vlacq, but to 10 places, and by Alexander John Thompson to 20 places in 1952.

Briggs was one of the first to use finite-difference methods to compute tables of functions. He also completed a table of logarithmic sines and tangents for the hundredth part of every degree to fourteen decimal places, with a table of natural sines to fifteen places and the tangents and secants for the same to ten places, all of which were printed at Gouda in 1631 and published in 1633 under the title of Trigonometria Britannica; this work was probably a successor to his 1617 Logarithmorum Chilias Prima ("The First Thousand Logarithms"), which gave a brief account of logarithms and a long table of the first 1000 integers calculated to the 14th decimal place.

Natural logarithm

Main article: Natural logarithm

 

Opus geometricum posthumum, 1668

In 1649, Alphonse Antonio de Sarasa, a former student of Grégoire de Saint-Vincent, related logarithms to the quadrature of the hyperbola, by pointing out that the area A(t) under the hyperbola from x = 1 to x = t satisfies

${\displaystyle A(tu)=A(t)+A(u).}$

At first the reaction to Saint-Vincent's hyperbolic logarithm was a continuation of studies of quadrature as in Christiaan Huygens (1651) and James Gregory (1667). Subsequently, an industry of making logarithms arose as "logaritmotechnia", the title of works by Nicholas Mercator (1668), Euclid Speidell (1688), and John Craig (1710)

By use of the geometric series with its conditional radius of convergence, an alternating series called the Mercator series expresses the logarithm function over the interval (0,2). Since the series is negative in (0,1), the "area under the hyperbola" must be considered negative there, so a signed measure, instead of purely positive area, determines the hyperbolic logarithm.

Historian Tom Whiteside described the transition to the analytic function as follows:

By the end of the 17th century we can say that much more than being a calculating device suitably well-tabulated, the logarithm function, very much on the model of the hyperbola-area, had been accepted into mathematics. When, in the 18th century, this geometric basis was discarded in favour of a fully analytical one, no extension or reformulation was necessary – the concept of "hyperbola-area" was transformed painlessly into "natural logarithm".

Leonard Euler treated a logarithm as an exponent of a certain number called the base of the logarithm. He noted that the number 2.71828, and its reciprocal, provided a point on the hyperbola xy = 1 such that an area of one square unit lies beneath the hyperbola, right of (1,1) and above the asymptote of the hyperbola. He then called the logarithm, with this number as base, the natural logarithm.

As noted by Howard Eves, "One of the anomalies in the history of mathematics is the fact that logarithms were discovered before exponents were in use." Carl B. Boyer wrote, "Euler was among the first to treat logarithms as exponents, in the manner now so familiar."

Pioneers of logarithms

Predecessors

The Babylonians sometime in 2000–1600 BC may have invented the quarter square multiplication algorithm to multiply two numbers using only addition, subtraction and a table of quarter squares. Thus, such a table served a similar purpose to tables of logarithms, which also allow multiplication to be calculated using addition and table lookups. However, the quarter-square method could not be used for division without an additional table of reciprocals (or the knowledge of a sufficiently simple algorithm to generate reciprocals). Large tables of quarter squares were used to simplify the accurate multiplication of large numbers from 1817 onwards until this was superseded by the use of computers.

The Indian mathematician Virasena worked with the concept of ardhaccheda: the number of times a number of the form 2n could be halved. For exact powers of 2, this equals the binary logarithm, but it differs from the logarithm for other numbers. He described a product formula for this concept and also introduced analogous concepts for base 3 (trakacheda) and base 4 (caturthacheda).

Michael Stifel published Arithmetica integra in Nuremberg in 1544, which contains a table of integers and powers of 2 that has been considered an early version of a table of binary logarithms.

In the 16th and early 17th centuries an algorithm called prosthaphaeresis was used to approximate multiplication and division. This used the trigonometric identity

${\displaystyle \cos \alpha \cos \beta =\frac{1}{2}[\cos (\alpha +\beta )+\cos (\alpha -\beta )]}$

or similar to convert the multiplications to additions and table lookups. However, logarithms are more straightforward and require less work. It can be shown using Euler's formula that the two techniques are related.

Ibn Hamza al-Maghribi

Ibn Hamza al-Maghribi, an Algerian mathematician, discovered logarithmic functions 23 years earlier, around 1591, with his work Âsâr-ı Bâkiye (literally in Turkish: The memories that remain).

Bürgi

The Swiss mathematician Jost Bürgi constructed a table of progressions which can be considered a table of antilogarithms independently of John Napier, whose publication (1614) was known by the time Bürgi published at the behest of Johannes Kepler. We know that Bürgi had some way of simplifying calculations around 1588, but most likely this way was the use of prosthaphaeresis, and not the use of his table of progressions which probably goes back to about 1600. Indeed, Wittich, who was in Kassel from 1584 to 1586, brought with him knowledge of prosthaphaeresis, a method by which multiplications and divisions can be replaced by additions and subtractions of trigonometrical values. This procedure achieves the same as the logarithms will a few years later.

Napier

John Napier (1550–1617), the inventor of logarithms

 

Main article: Napierian logarithm

The method of logarithms was first publicly propounded by John Napier in 1614, in a book titled Mirifici Logarithmorum Canonis Descriptio.

Johannes Kepler, who used logarithm tables extensively to compile his Ephemeris and therefore dedicated it to Napier, remarked:

... the accent in calculation led Justus Byrgius [Joost Bürgi] on the way to these very logarithms many years before Napier's system appeared; but ... instead of rearing up his child for the public benefit he deserted it in the birth.

— Johannes Kepler, Rudolphine Tables (1627)

Napier imagined a point P travelling across a line segment P0 to Q. Starting at P0, with a certain initial speed, P travels at a speed proportional to its distance to Q, causing P to never reach Q. Napier juxtaposed this figure with that of a point L travelling along an unbounded line segment, starting at L0, and with a constant speed equal to that of the initial speed of point P. Napier defined the distance from L0 to L as the logarithm of the distance from P to Q.

By repeated subtractions Napier calculated (1 − 10⁻⁷)^L for L ranging from 1 to 100. The result for L=100 is approximately 0.99999 = 1 − 10⁻⁵. Napier then calculated the products of these numbers with 10⁷(1 − 10⁻⁵)^L for L from 1 to 50, and did similarly with 0.9998 ≈ (1 − 10⁻⁵)²⁰ and 0.9 ≈ 0.995²⁰. These computations, which occupied 20 years, allowed him to give, for any number N from 5 to 10 million, the number L that solves the equation

${\displaystyle N={10}^7(1-{10}^{-7}{)}^L.}$

Napier first called L an "artificial number", but later introduced the word "logarithm" to mean a number that indicates a ratio: λόγος (logos) meaning proportion, and ἀριθμός (arithmos) meaning number. In modern notation, the relation to natural logarithms is: 

${\displaystyle L={\log }_{(1-{10}^{-7})}\left(\frac{N}{{10}^7}\right)\approx {10}^7{\log }_{1/e}\left(\frac{N}{{10}^7}\right)=-{10}^7{\log }_e\left(\frac{N}{{10}^7}\right),}$

where the very close approximation corresponds to the observation that

${\displaystyle (1-{10}^{-7}{)}^{{10}^7}\approx \frac{1}{e}.}$

The invention was quickly and widely met with acclaim. The works of Bonaventura Cavalieri (Italy), Edmund Wingate (France), Xue Fengzuo (China), and Johannes Kepler's Chilias logarithmorum (Germany) helped spread the concept further.

Euler

Graph of the equation y = 1/x. Here, Euler's number e makes the shaded area equal to 1.

Around 1730, Leonhard Euler defined the exponential function and the natural logarithm by

${\displaystyle \begin{array}{rl}{e}^x& =\underset{n\to \mathrm{\infty }}{lim}{\left(1+\frac{x}{n}\right)}^n,\\ \ln (x)& =\underset{n\to \mathrm{\infty }}{lim}n(x^{1/n}-1).\end{array}}$

In his 1748 textbook Introduction to the Analysis of the Infinite, Euler published the now-standard approach to logarithms via an inverse function: In chapter 6, "On exponentials and logarithms", he begins with a constant base a and discusses the transcendental function ${\displaystyle y=a^z.}$ Then its inverse is the logarithm:

z = log_a y.

Tables of logarithms

A page from Henry Briggs' 1617 Logarithmorum Chilias Prima showing the base-10 (common) logarithm of the integers 1 to 67 to fourteen decimal places.

 

Part of a 20th-century table of common logarithms in the reference book Abramowitz and Stegun.

 

A page from a table of logarithms of trigonometric functions from the 2002 American Practical Navigator. Columns of differences are included to aid interpolation.

Mathematical tables containing common logarithms (base-10) were extensively used in computations prior to the advent of computers and calculators, not only because logarithms convert problems of multiplication and division into much easier addition and subtraction problems, but for an additional property that is unique to base-10 and proves useful: Any positive number can be expressed as the product of a number from the interval [1,10) and an integer power of 10. This can be envisioned as shifting the decimal separator of the given number to the left yielding a positive, and to the right yielding a negative exponent of 10. Only the logarithms of these normalized numbers (approximated by a certain number of digits), which are called mantissas, need to be tabulated in lists to a similar precision (a similar number of digits). These mantissas are all positive and enclosed in the interval [0,1). The common logarithm of any given positive number is then obtained by adding its mantissa to the common logarithm of the second factor. This logarithm is called the characteristic of the given number. Since the common logarithm of a power of 10 is exactly the exponent, the characteristic is an integer number, which makes the common logarithm exceptionally useful in dealing with decimal numbers. For numbers less than 1, the characteristic makes the resulting logarithm negative, as required. See common logarithm for details on the use of characteristics and mantissas.

Early tables

Michael Stifel published Arithmetica integra in Nuremberg in 1544 which contains a table of integers and powers of 2 that has been considered an early version of a logarithmic table.

The first published table of logarithms was in John Napier's 1614, Mirifici Logarithmorum Canonis Descriptio. The book contained fifty-seven pages of explanatory matter and ninety pages of tables of trigonometric functions and their natural logarithms.

The English mathematician Henry Briggs visited Napier in 1615, and proposed a re-scaling of Napier's logarithms to form what is now known as the common or base-10 logarithms. Napier delegated to Briggs the computation of a revised table, and they later published, in 1617, Logarithmorum Chilias Prima ("The First Thousand Logarithms"), which gave a brief account of logarithms and a table for the first 1000 integers calculated to the 14th decimal place.

In 1624, Briggs' Arithmetica Logarithmica appeared in folio as a work containing the logarithms of 30,000 natural numbers to fourteen decimal places (1-20,000 and 90,001 to 100,000). This table was later extended by Adriaan Vlacq, but to 10 places, and by Alexander John Thompson to 20 places in 1952.

Briggs was one of the first to use finite-difference methods to compute tables of functions.

Vlacq's table was later found to contain 603 errors, but "this cannot be regarded as a great number, when it is considered that the table was the result of an original calculation, and that more than 2,100,000 printed figures are liable to error." An edition of Vlacq's work, containing many corrections, was issued at Leipzig in 1794 under the title Thesaurus Logarithmorum Completus by Jurij Vega.

François Callet's seven-place table (Paris, 1795), instead of stopping at 100,000, gave the eight-place logarithms of the numbers between 100,000 and 108,000, in order to diminish the errors of interpolation, which were greatest in the early part of the table, and this addition was generally included in seven-place tables. The only important published extension of Vlacq's table was made by Edward Sang in 1871, whose table contained the seven-place logarithms of all numbers below 200,000.

Briggs and Vlacq also published original tables of the logarithms of the trigonometric functions. Briggs completed a table of logarithmic sines and logarithmic tangents for the hundredth part of every degree to fourteen decimal places, with a table of natural sines to fifteen places and the tangents and secants for the same to ten places, all of which were printed at Gouda in 1631 and published in 1633 under the title of Trigonometria Britannica. Tables logarithms of trigonometric functions simplify hand calculations where a function of an angle must be multiplied by another number, as is often the case.

Besides the tables mentioned above, a great collection, called Tables du Cadastre, was constructed under the direction of Gaspard de Prony, by an original computation, under the auspices of the French republican government of the 1790s. This work, which contained the logarithms of all numbers up to 100,000 to nineteen places, and of the numbers between 100,000 and 200,000 to twenty-four places, exists only in manuscript, "in seventeen enormous folios," at the Observatory of Paris. It was begun in 1792, and "the whole of the calculations, which to secure greater accuracy were performed in duplicate, and the two manuscripts subsequently collated with care, were completed in the short space of two years." Cubic interpolation could be used to find the logarithm of any number to a similar accuracy.

For different needs, logarithm tables ranging from small handbooks to multi-volume editions have been compiled:

Year	Author	Range	Decimal places	Note
1614	John Napier, Mirifici Logarithmorum Canonis Descriptio	0°–90°, by minutes	7	sin(Θ) & ln(sin(Θ)), see image
1617	Henry Briggs, Logarithmorum Chilias Prima	1–1000	14	see image
1624	Henry Briggs Arithmetica Logarithmica	1–20,000, 90,000–100,000	14
1628	Adriaan Vlacq	20,000–90,000	10	contained only 603 errors
1792–94	Gaspard de Prony Tables du Cadastre	1–100,000 and 100,000–200,000	19 and 24, respectively	"seventeen enormous folios", never published
1794	Jurij Vega Thesaurus Logarithmorum Completus (Leipzig)			corrected edition of Vlacq's work
1795	François Callet (Paris)	100,000–108,000	7
1871	Edward Sang	1–200,000	7

Slide rule

Main article: Slide rule

 

William Oughtred (1575–1660), inventor of the circular slide rule.

 

A collection of slide rules at the Museum of the History of Science, Oxford

The slide rule was invented around 1620–1630, shortly after John Napier's publication of the concept of the logarithm. Edmund Gunter of Oxford developed a calculating device with a single logarithmic scale; with additional measuring tools it could be used to multiply and divide. The first description of this scale was published in Paris in 1624 by Edmund Wingate (c.1593–1656), an English mathematician, in a book entitled L'usage de la reigle de proportion en l'arithmetique & geometrie. The book contains a double scale, logarithmic on one side, tabular on the other. In 1630, William Oughtred of Cambridge invented a circular slide rule, and in 1632 combined two handheld Gunter rules to make a device that is recognizably the modern slide rule. Like his contemporary at Cambridge, Isaac Newton, Oughtred taught his ideas privately to his students. Also like Newton, he became involved in a vitriolic controversy over priority, with his one-time student Richard Delamain and the prior claims of Wingate. Oughtred's ideas were only made public in publications of his student William Forster in 1632 and 1653.

In 1677, Henry Coggeshall created a two-foot folding rule for timber measure, called the Coggeshall slide rule, expanding the slide rule's use beyond mathematical inquiry.

In 1722, Warner introduced the two- and three-decade scales, and in 1755 Everard included an inverted scale; a slide rule containing all of these scales is usually known as a "polyphase" rule.

In 1815, Peter Mark Roget invented the log log slide rule, which included a scale displaying the logarithm of the logarithm. This allowed the user to directly perform calculations involving roots and exponents. This was especially useful for fractional powers.

In 1821, Nathaniel Bowditch, described in the American Practical Navigator a "sliding rule" that contained scales trigonometric functions on the fixed part and a line of log-sines and log-tans on the slider used to solve navigation problems.

In 1845, Paul Cameron of Glasgow introduced a Nautical Slide-Rule capable of answering navigation questions, including right ascension and declination of the sun and principal stars.

Modern form

Engineer using a slide rule, with mechanical calculator in background, mid 20th century

A more modern form of slide rule was created in 1859 by French artillery lieutenant Amédée Mannheim, "who was fortunate in having his rule made by a firm of national reputation and in having it adopted by the French Artillery." It was around this time that engineering became a recognized profession, resulting in widespread slide rule use in Europe–but not in the United States. There Edwin Thacher's cylindrical rule took hold after 1881. The duplex rule was invented by William Cox in 1891, and was produced by Keuffel and Esser Co. of New York.

Impact

Writing in 1914 on the 300th anniversary of Napier's tables, E. W. Hobson described logarithms as "providing a great labour-saving instrument for the use of all those who have occasion to carry out extensive numerical calculations" and comparing it in importance to the "Indian invention" of our decimal number system. Napier's improved method of calculation was soon adopted in Britain and Europe. Kepler dedicated his 1620 Ephereris to Napier, congratulating him on his invention and its benefits to astronomy. Edward Wright, an authority on celestial navigation, translated Napier's Latin Descriptio into English in 1615, shortly after its publication. Briggs extended the concept to the more convenient base 10, or common logarithm.

“Probably no work has ever influenced science as a whole, and mathematics in particular, so profoundly as this modest little book [the Descriptio]. It opened the way for the abolition, once and for all, of the infinitely laborious, nay, nightmarish, processes of long division and multiplication, of finding the power and the root of numbers.”

The logarithm function remains a staple of mathematical analysis, but printed tables of logarithms gradually diminished in importance in the twentieth century as mechanical calculators and, later, electronic computers took over computations that required high accuracy. The introduction of hand-held scientific calculators in the 1970s ended the era of slide rules. Logarithmic scale graphs are widely used to display data with a wide range. The decibel, a logarithmic unit, is also widely used. The current, 2002, edition of The American Practical Navigator (Bowditch) still contains tables of logarithms and logarithms of trigonometric functions.

Natural logarithm

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Natural_logarithm

 

Natural logarithm
Graph of part of the natural logarithm function. The function slowly grows to positive infinity as x increases, and slowly goes to negative infinity as x approaches 0 ("slowly" as compared to any power law of x).
General information
General definition	${\displaystyle \ln x={\log }_{e}x}$
Motivation of invention	Analytic proofs
Fields of application	Pure and applied mathematics
Domain, Codomain and Image
Domain	${\displaystyle {\mathbb{R}}_{>0}}$
Codomain	${\displaystyle \mathbb{R}}$
Image	${\displaystyle \mathbb{R}}$
Specific values
Value at +∞	+∞
Value at e	1
Specific features
Asymptote	${\displaystyle x=0}$
Root	1
Inverse	${\displaystyle \exp x}$
Derivative	${\displaystyle {\displaystyle \frac{d}{dx}}\ln x={\displaystyle \frac{1}{x}},x>0}$
Antiderivative	${\displaystyle \int \ln xdx=x\left(\ln x-1\right)+C}$

The natural logarithm of a number is its logarithm to the base of the mathematical constant e, which is an irrational and transcendental number approximately equal to 2.718281828459. The natural logarithm of x is generally written as ln x, log_e x, or sometimes, if the base e is implicit, simply log x. Parentheses are sometimes added for clarity, giving ln(x), log_e(x), or log(x). This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity.

The natural logarithm of x is the power to which e would have to be raised to equal x. For example, ln 7.5 is 2.0149..., because e^2.0149... = 7.5. The natural logarithm of e itself, ln e, is 1, because e¹ = e, while the natural logarithm of 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 (with the area being negative when 0 < a < 1). The simplicity of this definition, which is matched in many other formulas involving the natural logarithm, leads to the term "natural". The definition of the natural logarithm can then be extended to give logarithm values for negative numbers and for all non-zero complex numbers, although this leads to a multi-valued function: see Complex logarithm for more.

The natural logarithm function, if considered as a real-valued function of a positive real variable, is the inverse function of the exponential function, leading to the identities:

${\displaystyle \begin{array}{rl}{e}^{\ln x}& =x\text{ if }x\in {\mathbb{R}}_{+}\\ \ln e^x& =x\text{ if }x\in \mathbb{R}\end{array}}$

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

${\displaystyle \ln (x\cdot y)=\ln x+\ln y.}$

Logarithms can be defined for any positive base other than 1, not only e. However, logarithms in other bases differ only by a constant multiplier from the natural logarithm, and can be defined in terms of the latter, ${\displaystyle {\log }_{b}x=\ln x/\ln b=\ln x\cdot {\log }_{b}e}$.

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 scientific disciplines, and are used to solve problems involving compound interest.

History

Main article: History of logarithms

The concept of the natural logarithm was worked out by Gregoire de Saint-Vincent and Alphonse Antonio de Sarasa before 1649. Their work involved quadrature of the hyperbola with equation xy = 1, by determination of the area of hyperbolic sectors. Their solution generated the requisite "hyperbolic logarithm" function, which had the properties now associated with the natural logarithm.

An early mention of the natural logarithm was by Nicholas Mercator in his work Logarithmotechnia, published in 1668, although the mathematics teacher John Speidell had already compiled a table of what in fact were effectively natural logarithms in 1619. It has been said that Speidell's logarithms were to the base e, but this is not entirely true due to complications with the values being expressed as integers.

Notational conventions

The notations ln x and log_e x both refer unambiguously to the natural logarithm of x, and log x without an explicit base may also refer to the natural logarithm. This usage is common in mathematics, along with some scientific contexts as well as in many programming languages. In some other contexts such as chemistry, however, log x can be used to denote the common (base 10) logarithm. It may also refer to the binary (base 2) logarithm in the context of computer science, particularly in the context of time complexity.

Definitions

The natural logarithm can be defined in several equivalent ways.

Inverse of exponential

The most general definition is as the inverse function of ${\displaystyle e^x}$, so that ${\displaystyle e^{\ln (x)}=x}$. Because ${\displaystyle e^x}$ is positive and invertible for any real input ${\displaystyle x}$, this definition of ${\displaystyle \ln (x)}$ is well defined for any positive x. For the complex numbers, ${\displaystyle e^z}$ is not invertible, so ${\displaystyle \ln (z)}$ is a multivalued function. In order to make ${\displaystyle \ln (z)}$ a proper, single-output function, we therefore need to restrict it to a particular principal branch, often denoted by ${\displaystyle \mathrm{Ln}(z)}$. As the inverse function of ${\displaystyle e^z}$, ${\displaystyle \ln (z)}$ can be defined by inverting the usual definition of ${\displaystyle e^z}$:

${\displaystyle e^z=\underset{n\to \mathrm{\infty }}{lim}{\left(1+\frac{z}{n}\right)}^n}$

Doing so yields:

${\displaystyle \ln (z)=\underset{n\to \mathrm{\infty }}{lim}n\cdot (\sqrt[n]{z}-1)}$

This definition therefore derives its own principal branch from the principal branch of nth roots.

Integral definition

ln a as the area of the shaded region under the curve f(x) = 1/x from 1 to a. If a is less than 1, the area taken to be negative.

 

The area under the hyperbola satisfies the logarithm rule. Here A(s,t) denotes the area under the hyperbola between s and t.

The natural logarithm of a positive, real number a may be defined as the area under the graph of the hyperbola with equation y = 1/x between x = 1 and x = a. This is the integral

${\displaystyle \ln a={\int }_1^a\frac{1}{x}dx.}$

If a is less than 1, then this area is considered to be negative.

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

${\displaystyle \ln (ab)=\ln a+\ln b.}$

This can be demonstrated by splitting the integral that defines ln ab into two parts, and then making the variable substitution x = at (so dx = a dt) in the second part, as follows:

${\displaystyle \begin{array}{rl}\ln ab={\int }_1^{ab}\frac{1}{x}dx& ={\int }_1^a\frac{1}{x}dx+{\int }_a^{ab}\frac{1}{x}dx\\ & ={\int }_1^a\frac{1}{x}dx+{\int }_1^b\frac{1}{at}adt\\ & ={\int }_1^a\frac{1}{x}dx+{\int }_1^b\frac{1}{t}dt\\ & =\ln a+\ln b.\end{array}}$

In elementary terms, this is simply scaling by 1/a in the horizontal direction and by a in the vertical direction. Area does not change under this transformation, but the region between a and ab is reconfigured. Because the function a/(ax) is equal to the function 1/x, the resulting area is precisely ln b.

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

The natural logarithm also has an improper integral representation, which can be derived with Fubini's theorem as follows:

${\displaystyle \ln \left(x\right)={\int }_1^x\frac{1}{u}du={\int }_1^x{\int }_0^{\mathrm{\infty }}{e}^{-tu}dtdu={\int }_0^{\mathrm{\infty }}{\int }_1^x{e}^{-tu}dudt={\int }_0^{\mathrm{\infty }}\frac{e^{-t}-e^{-tx}}{t}dt}$

Properties

• ${\displaystyle \ln 1=0}$
• ${\displaystyle \ln e=1}$
• ${\displaystyle \ln (xy)=\ln x+\ln y\text{for }x>0\text{and }y>0}$
• ${\displaystyle \ln (x/y)=\ln x-\ln y}$
• ${\displaystyle \ln (x^y)=y\ln x\text{for }x>0}$
• ${\displaystyle \ln x<\ln y\text{for }0<x<y}$
• ${\displaystyle \underset{x\to 0}{lim}\frac{\ln (1+x)}{x}=1}$
• ${\displaystyle \underset{\alpha \to 0}{lim}\frac{x^{\alpha }-1}{\alpha }=\ln x\text{for }x>0}$
• ${\displaystyle \frac{x-1}{x}\le \ln x\le x-1\text{for}x>0}$
• ${\displaystyle \ln (1+x^{\alpha })\le \alpha x\text{for}x\ge 0\text{and }\alpha \ge 1}$

Derivative

The derivative of the natural logarithm as a real-valued function on the positive reals is given by

${\displaystyle \frac{d}{dx}\ln x=\frac{1}{x}.}$

How to establish this derivative of the natural logarithm depends on how it is defined firsthand. If the natural logarithm is defined as the integral

${\displaystyle \ln x={\int }_1^x\frac{1}{t}dt,}$

then the derivative immediately follows from the first part of the fundamental theorem of calculus.

On the other hand, if the natural logarithm is defined as the inverse of the (natural) exponential function, then the derivative (for x > 0) can be found by using the properties of the logarithm and a definition of the exponential function. From the definition of the number ${\displaystyle e=\underset{u\to 0}{lim}(1+u{)}^{1/u},}$ the exponential function can be defined as ${\displaystyle e^x=\underset{u\to 0}{lim}(1+u{)}^{x/u}=\underset{h\to 0}{lim}(1+hx{)}^{1/h}}$, where ${\displaystyle u=hx,h=u/x.}$ The derivative can then be found from first principles.

${\displaystyle \begin{array}{rl}\frac{d}{dx}\ln x& =\underset{h\to 0}{lim}\frac{\ln (x+h)-\ln x}{h}\\ & =\underset{h\to 0}{lim}\left[\frac{1}{h}\ln \left(\frac{x+h}{x}\right)\right]\\ & =\underset{h\to 0}{lim}\left[\ln \left({\left(1+\frac{h}{x}\right)}^{\frac{1}{h}}\right)\right]& & \text{all above for logarithmic properties}\\ & =\ln \left[\underset{h\to 0}{lim}{\left(1+\frac{h}{x}\right)}^{\frac{1}{h}}\right]& & \text{for continuity of the logarithm}\\ & =\ln e^{1/x}& & \text{for the definition of }{e}^x=\underset{h\to 0}{lim}(1+hx{)}^{1/h}\\ & =\frac{1}{x}& & \text{for the definition of the ln as inverse function.}\end{array}}$

Also, we have:

${\displaystyle \frac{d}{dx}\ln ax=\frac{d}{dx}(\ln a+\ln x)=\frac{d}{dx}\ln a+\frac{d}{dx}\ln x=\frac{1}{x}.}$

so, unlike its inverse function ${\displaystyle e^{ax}}$, a constant in the function doesn't alter the differential.

Series

The Taylor polynomials for ln(1 + x) only provide accurate approximations in the range −1 < x ≤ 1. Beyond some x > 1, the Taylor polynomials of higher degree are increasingly worse approximations.

Since the natural logarithm is undefined at 0, ${\displaystyle \ln (x)}$ itself does not have a Maclaurin series, unlike many other elementary functions. Instead, one looks for Taylor expansions around other points. For example, if ${\displaystyle |x-1|\le 1\text{ and }x\ne 0,}$ then

${\displaystyle \begin{array}{rl}\ln x& ={\int }_1^x\frac{1}{t}dt={\int }_0^{x-1}\frac{1}{1+u}du\\ & ={\int }_0^{x-1}(1-u+u^2-u^3+\cdots )du\\ & =(x-1)-\frac{(x-1{)}^2}{2}+\frac{(x-1{)}^3}{3}-\frac{(x-1{)}^4}{4}+\cdots \\ & =\sum _{k=1}^{\mathrm{\infty }}\frac{(-1{)}^{k-1}(x-1{)}^k}{k}.\end{array}}$

This is the Taylor series for ln x around 1. A change of variables yields the Mercator series:

${\displaystyle \ln (1+x)=\sum _{k=1}^{\mathrm{\infty }}\frac{(-1{)}^{k-1}}{k}{x}^k=x-\frac{x^2}{2}+\frac{x^3}{3}-\cdots ,}$

valid for |x| ≤ 1 and x ≠ −1.

Leonhard Euler, disregarding ${\displaystyle x\ne -1}$, nevertheless applied this series to x = −1 to show that the harmonic series equals the natural logarithm of 1/(1 − 1), that is, the logarithm of infinity. Nowadays, more formally, one can prove that the harmonic series truncated at N is close to the logarithm of N, when N is large, with the difference converging to the Euler–Mascheroni constant.

The figure is a graph of ln(1 + x) and some of its Taylor polynomials around 0. These approximations converge to the function only in the region −1 < x ≤ 1; outside this region, the higher-degree Taylor polynomials devolve to worse approximations for the function.

A useful special case for positive integers n, taking ${\displaystyle x=\frac{1}{n}}$, is:

${\displaystyle \ln \left(\frac{n+1}{n}\right)=\sum _{k=1}^{\mathrm{\infty }}\frac{(-1{)}^{k-1}}{k{n}^k}=\frac{1}{n}-\frac{1}{2n^2}+\frac{1}{3n^3}-\frac{1}{4n^4}+\cdots }$

If ${\displaystyle \mathrm{Re}(x)\ge 1/2,}$ then

${\displaystyle \begin{array}{rl}\ln (x)& =-\ln \left(\frac{1}{x}\right)=-\sum _{k=1}^{\mathrm{\infty }}\frac{(-1{)}^{k-1}(\frac{1}{x}-1{)}^k}{k}=\sum _{k=1}^{\mathrm{\infty }}\frac{(x-1{)}^k}{k{x}^k}\\ & =\frac{x-1}{x}+\frac{(x-1{)}^2}{2x^2}+\frac{(x-1{)}^3}{3x^3}+\frac{(x-1{)}^4}{4x^4}+\cdots \end{array}}$

Now, taking ${\displaystyle x=\frac{n+1}{n}}$ for positive integers n, we get:

${\displaystyle \ln \left(\frac{n+1}{n}\right)=\sum _{k=1}^{\mathrm{\infty }}\frac{1}{k(n+1{)}^k}=\frac{1}{n+1}+\frac{1}{2(n+1{)}^2}+\frac{1}{3(n+1{)}^3}+\frac{1}{4(n+1{)}^4}+\cdots }$

If ${\displaystyle \mathrm{Re}(x)\ge 0\text{ and }x\ne 0,}$ then

${\displaystyle \ln (x)=\ln \left(\frac{2x}{2}\right)=\ln \left(\frac{1+\frac{x-1}{x+1}}{1-\frac{x-1}{x+1}}\right)=\ln \left(1+\frac{x-1}{x+1}\right)-\ln \left(1-\frac{x-1}{x+1}\right).}$

Since

${\displaystyle \begin{array}{rl}\ln (1+y)-\ln (1-y)& =\sum _{i=1}^{\mathrm{\infty }}\frac{1}{i}\left((-1{)}^{i-1}{y}^i-(-1{)}^{i-1}(-y{)}^i\right)=\sum _{i=1}^{\mathrm{\infty }}\frac{y^i}{i}\left((-1{)}^{i-1}+1\right)\\ & =y\sum _{i=1}^{\mathrm{\infty }}\frac{y^{i-1}}{i}\left((-1{)}^{i-1}+1\right)\stackrel{i-1\to 2k}{=}2y\sum _{k=0}^{\mathrm{\infty }}\frac{y^{2k}}{2k+1},\end{array}}$

we arrive at

${\displaystyle \begin{array}{rl}\ln (x)& =\frac{2(x-1)}{x+1}\sum _{k=0}^{\mathrm{\infty }}\frac{1}{2k+1}{\left(\frac{(x-1{)}^2}{(x+1{)}^2}\right)}^k\\ & =\frac{2(x-1)}{x+1}\left(\frac{1}{1}+\frac{1}{3}\frac{(x-1{)}^2}{(x+1{)}^2}+\frac{1}{5}{\left(\frac{(x-1{)}^2}{(x+1{)}^2}\right)}^2+\cdots \right).\end{array}}$

Using the substitution ${\displaystyle x=\frac{n+1}{n}}$ again for positive integers n, we get:

${\displaystyle \begin{array}{rl}\ln \left(\frac{n+1}{n}\right)& =\frac{2}{2n+1}\sum _{k=0}^{\mathrm{\infty }}\frac{1}{(2k+1)((2n+1{)}^2{)}^k}\\ & =2\left(\frac{1}{2n+1}+\frac{1}{3(2n+1{)}^3}+\frac{1}{5(2n+1{)}^5}+\cdots \right).\end{array}}$

This is, by far, the fastest converging of the series described here.

The natural logarithm can also be expressed as an infinite product:

${\displaystyle \ln (x)=(x-1)\prod _{k=1}^{\mathrm{\infty }}\left(\frac{2}{1+\sqrt[2^k]{x}}\right)}$

Two examples might be:

${\displaystyle \ln (2)=\left(\frac{2}{1+\sqrt{2}}\right)\left(\frac{2}{1+\sqrt[4]{2}}\right)\left(\frac{2}{1+\sqrt[8]{2}}\right)\left(\frac{2}{1+\sqrt[16]{2}}\right)...}$

${\displaystyle \pi =(2i+2)\left(\frac{2}{1+\sqrt{i}}\right)\left(\frac{2}{1+\sqrt[4]{i}}\right)\left(\frac{2}{1+\sqrt[8]{i}}\right)\left(\frac{2}{1+\sqrt[16]{i}}\right)...}$

From this identity, we can easily get that:

${\displaystyle \frac{1}{\ln (x)}=\frac{x}{x-1}-\sum _{k=1}^{\mathrm{\infty }}\frac{2^{-k}{x}^{2^{-k}}}{1+x^{2^{-k}}}}$

For example:

${\displaystyle \frac{1}{\ln (2)}=2-\frac{\sqrt{2}}{2+2\sqrt{2}}-\frac{\sqrt[4]{2}}{4+4\sqrt[4]{2}}-\frac{\sqrt[8]{2}}{8+8\sqrt[8]{2}}\cdots }$

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:

${\displaystyle \frac{d}{dx}\ln \left|x\right|=\frac{1}{x},x\ne 0}$

In other words, when integrating over an interval of the real line that does not include ${\displaystyle x=0}$ then

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

where C is an arbitrary constant of integration. Likewise, when the integral is over an interval where ${\displaystyle f(x)\ne 0}$,

${\displaystyle \int \frac{f^{\prime }(x)}{f(x)}dx=\ln |f(x)|+C.}$

For example, consider the integral of tan(x) over an interval that does not include points where tan(x) is infinite:

${\displaystyle \int \tan xdx=-\int \frac{\frac{d}{dx}\cos x}{\cos x}dx=-\ln \left|\cos x\right|+C=\ln \left|\sec x\right|+C.}$

The natural logarithm can be integrated using integration by parts:

${\displaystyle \int \ln xdx=x\ln x-x+C.}$

Let:

${\displaystyle u=\ln x\Rightarrow du=\frac{dx}{x}}$

${\displaystyle dv=dx\Rightarrow v=x}$

then:

${\displaystyle \begin{array}{rl}\int \ln xdx& =x\ln x-\int \frac{x}{x}dx\\ & =x\ln x-\int 1dx\\ & =x\ln x-x+C\end{array}}$

Efficient computation

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

${\displaystyle \begin{array}{rl}\ln 123.456& =\ln (1.23456\cdot {10}^2)\\ & =\ln 1.23456+\ln ({10}^2)\\ & =\ln 1.23456+2\ln 10\\ & \approx \ln 1.23456+2\cdot \mathrm{2.3025851.}\end{array}}$

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

Natural logarithm of 10

The natural logarithm of 10, which has the decimal expansion 2.30258509..., plays a role for example in the computation of natural logarithms of numbers represented in scientific notation, as a mantissa multiplied by a power of 10:

${\displaystyle \ln (a\cdot {10}^n)=\ln a+n\ln 10.}$

This means that one can effectively calculate the logarithms of numbers with very large or very small magnitude using the logarithms of a relatively small set of decimals in the range [1, 10).

High precision

To compute the natural logarithm with many digits of precision, the Taylor series approach is not efficient since the convergence is slow. Especially if x is near 1, a good alternative is to use Halley's method or Newton's method to invert the exponential function, because the series of the exponential function converges more quickly. For finding the value of y to give exp(y) − x = 0 using Halley's method, or equivalently to give exp(y/2) − x exp(−y/2) = 0 using Newton's method, the iteration simplifies to

${\displaystyle y_{n+1}=y_n+2\cdot \frac{x-\exp (y_n)}{x+\exp (y_n)}}$

which has cubic convergence to ln(x).

Another alternative for extremely high precision calculation is the formula

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

where M denotes the arithmetic-geometric mean of 1 and 4/s, and

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

with m chosen so that p bits of precision is attained. (For most purposes, the value of 8 for m is sufficient.) 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.) Or, the following formula can be used:

${\displaystyle \ln x=\frac{\pi }{M\left({\theta }_2^2(1/x),{\theta }_3^2(1/x)\right)},x\in (1,\mathrm{\infty })}$

where

${\displaystyle {\theta }_2(x)=\sum _{n\in \mathbb{Z}}{x}^{(n+1/2{)}^2},{\theta }_3(x)=\sum _{n\in \mathbb{Z}}{x}^{n^2}}$

are the Jacobi theta functions.

Based on a proposal by William Kahan and first implemented in the Hewlett-Packard HP-41C calculator in 1979 (referred to under "LN1" in the display, only), some calculators, operating systems (for example Berkeley UNIX 4.3BSD), computer algebra systems and programming languages (for example C99) provide a special natural logarithm plus 1 function, alternatively named LNP1, or log1p to give more accurate results for logarithms close to zero by passing arguments x, also close to zero, to a function log1p(x), which returns the value ln(1+x), instead of passing a value y close to 1 to a function returning ln(y). The function log1p avoids in the floating point arithmetic a near cancelling of the absolute term 1 with the second term from the Taylor expansion of the ln. This keeps the argument, the result, and intermediate steps all close to zero where they can be most accurately represented as floating-point numbers.

In addition to base e the IEEE 754-2008 standard defines similar logarithmic functions near 1 for binary and decimal logarithms: log₂(1 + x) and log₁₀(1 + x).

Similar inverse functions named "expm1", "expm" or "exp1m" exist as well, all with the meaning of expm1(x) = exp(x) − 1.

An identity in terms of the inverse hyperbolic tangent,

${\displaystyle \mathrm{l}\mathrm{o}\mathrm{g}1\mathrm{p}(x)=\log (1+x)=2\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\frac{x}{2+x}\right),}$

gives a high precision value for small values of x on systems that do not implement log1p(x).

Computational complexity

Main article: Computational complexity of mathematical operations

The computational complexity of computing the natural logarithm using the arithmetic-geometric mean (for both of the above methods) 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.

Continued fractions

While no simple continued fractions are available, several generalized continued fractions are, including:

${\displaystyle \begin{array}{rl}\ln (1+x)& =\frac{x^1}{1}-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\frac{x^5}{5}-\cdots \\ & =\frac{x}{1-0x+\frac{1^{2}x}{2-1x+\frac{2^{2}x}{3-2x+\frac{3^{2}x}{4-3x+\frac{4^{2}x}{5-4x+\ddots }}}}}\end{array}}$

${\displaystyle \begin{array}{rl}\ln \left(1+\frac{x}{y}\right)& =\frac{x}{y+\frac{1x}{2+\frac{1x}{3y+\frac{2x}{2+\frac{2x}{5y+\frac{3x}{2+\ddots }}}}}}\\ & =\frac{2x}{2y+x-\frac{(1x{)}^2}{3(2y+x)-\frac{(2x{)}^2}{5(2y+x)-\frac{(3x{)}^2}{7(2y+x)-\ddots }}}}\end{array}}$

These continued fractions—particularly the last—converge rapidly for values close to 1. However, the natural logarithms of much larger numbers can easily be computed, by repeatedly adding those of smaller numbers, with similarly rapid convergence.

For example, since 2 = 1.25³ × 1.024, the natural logarithm of 2 can be computed as:

${\displaystyle \begin{array}{rl}\ln 2& =3\ln \left(1+\frac{1}{4}\right)+\ln \left(1+\frac{3}{125}\right)\\ & =\frac{6}{9-\frac{1^2}{27-\frac{2^2}{45-\frac{3^2}{63-\ddots }}}}+\frac{6}{253-\frac{3^2}{759-\frac{6^2}{1265-\frac{9^2}{1771-\ddots }}}}.\end{array}}$

Furthermore, since 10 = 1.25¹⁰ × 1.024³, even the natural logarithm of 10 can be computed similarly as:

${\displaystyle \begin{array}{rl}\ln 10& =10\ln \left(1+\frac{1}{4}\right)+3\ln \left(1+\frac{3}{125}\right)\\ & =\frac{20}{9-\frac{1^2}{27-\frac{2^2}{45-\frac{3^2}{63-\ddots }}}}+\frac{18}{253-\frac{3^2}{759-\frac{6^2}{1265-\frac{9^2}{1771-\ddots }}}}.\end{array}}$

The reciprocal of the natural logarithm can be also written in this way:

${\displaystyle \frac{1}{\ln (x)}=\frac{2x}{x^2-1}\sqrt{\frac{1}{2}+\frac{x^2+1}{4x}}\sqrt{\frac{1}{2}+\frac{1}{2}\sqrt{\frac{1}{2}+\frac{x^2+1}{4x}}}\dots }$

For example:

${\displaystyle \frac{1}{\ln (2)}=\frac{4}{3}\sqrt{\frac{1}{2}+\frac{5}{8}}\sqrt{\frac{1}{2}+\frac{1}{2}\sqrt{\frac{1}{2}+\frac{5}{8}}}\dots }$

Complex logarithms

Main article: Complex logarithm

The exponential function can be extended to a function which gives a complex number as e^z for any arbitrary complex number z; simply use the infinite series with x=z 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 e^x = 0; and it turns out that e^2iπ = 1 = e⁰. Since the multiplicative property still works for the complex exponential function, e^z = e^z+2kiπ, for all complex z and integers k.

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 2iπ at will. The complex logarithm can only be single-valued on the cut plane. For example, ln i = iπ/2 or 5iπ/2 or -3iπ/2, etc.; and although i⁴ = 1, 4 ln i can be defined as 2iπ, or 10iπ or −6iπ, and so on.

• Plots of the natural logarithm function on the complex plane (principal branch)
• 

  z = Re(ln(x + yi))

• 

  z = |(Im(ln(x + yi)))|

• 

  z = |(ln(x + yi))|

• 

  Superposition of the previous three graphs