Logarithm

In mathematics, the logarithm functions are the inverses of the exponential functions. Logarithms are numbers that are substituted in computation for other numbers, to which they bear such a relation that the operations to be performed on the latter are represented by simpler operations performed on the former. Logarithms convert multiplication to addition, division to subtraction (making them isomorphisms between the field operations), exponentiation to multiplication, and roots to division (making them crucial to slide rule construction).

History

The method of logarithms was first propounded in 1614, in a book entitled Mirifici Logarithmorum Canonis Descriptio, by John Napier (latinized Neperus), Baron of Merchiston in Scotland, who was born about 1550, and died in 1618, four years after the publication of his memorable invention. This method contributed to the advance of science, and especially of astronomy, by facilitating the difficult calculations without which that advance could not have been made. Prior to the advent of calculators and computers, it was constantly used in surveying, navigation, and other branches of practical mathematics. Besides their usefulness in computation, logarithms also fill an important place in the higher theoretical mathematics.

Joost Bürgi, a Swiss clockmaker in the employ of the Duke of Hesse-Kassel had, however, first conceived of logarithms, but only published later.

The word logarithm, which is due to Napier, is formed from λoγoς (logos), ratio, and αριθμoς (arithmos), number, and means a number that indicates a ratio. It refers to the proposition which was made by Napier his fundamental theorem, that the difference of two logarithms determines the ratio of the numbers for which they stand, so that an arithmetical series of logarithms corresponds to a geometric series of numbers.

Definitions

The base of a system of logarithms is a fixed number to which all numbers are referred in that system. It may have positive value except unity.

The logarithm of a number in any system is the exponent of the power to which the base of the system must be raised to produce that number.

The antilogarithm of a number is that number of which the given number is the logarithm; in other words, it is that power of the base of which the given number is the exponent.

So, in a system of logarithms of which 8 is the base,

$\log 8=1,\quad \mathrm {antilog} \,1=8$

$\log 64=2,\quad \mathrm {antilog} \,2=64$

$\log 512=3,\quad \mathrm {antilog} \,3=512$

$\log 4096=4,\quad \mathrm {antilog} \,4=4096$

$\cdots$

In general, if $x$ and $u$ have any such values as to satisfy the equation $a^{u}=x$, we have, in the system of which $a$ is the base, $u=\log x$ and $x=\mathrm {antilog} u$ and either of the two latter equations may be regarded as equivalent to the former.

Napier at first called logarithms artificial numbers, and antilogarithms natural numbers. The word antilogarithm was introduced in the 1800's and, while convenient, its use was never widespread.

Usage

Logarithms to various bases: red is to base e, green is to base 10, and purple is to base 1.7. Note how all basic logarithms pass through the point (1, 0).

If b > 0 and x = b^y, then y is the logarithm of x in the base b (meaning y is the power we have to raise b to, in order to get x), and we write log_bx = y. For example,

$\log _{10}100=2,\ {\mbox{since}}\ 10^{2}=100.$

$\log _{2}8=3,\ {\mbox{since}}\ 2^{3}=8.$

The function log_b(x) is defined whenever x is a positive real number and b is a positive real number different from 1. See logarithmic identities for several rules governing the logarithm functions.

Logarithms are useful in order to solve equations in which the unknown appears in the exponent, and they often occur as the solution of differential equations because of their simple derivatives. Furthermore, various quantities in science are expressed by their logarithms; see logarithmic scale for an explanation and a list.

For integers a and b, the number log_ba is irrational (i.e., not a quotient of two integers) if one of a and b has a prime factor which the other does not (and in particular if they are coprime and both greater than 1).

Common, or denary, logarithms

Main article: common logarithm.

The denary', or decimal, system is that which has the number 10 for its base. In practical computation without electronic calculators or the like, it is more convenient than any other system because of its relation to the ordinary numeral system of arithmetic. For details, including the roles of characteristics and mantissas of base 10 logarithms in computations, see common logarithm. It was proposed in 1617 by Henry Briggs, then Professor of Geometry in Gresham College, London, and afterwards Savilian Professor of Geometry at Oxford University. Briggs was among the first to recognize the importance of the invention of logarithms, and he made two journeys to Scotland for the purpose of visiting Napier, in consultation with whom he formed his new system - a system distinguished from the original one not only in being founded on the number 10, but also in some important simplifications of theory.

Historically, denary logarithms were sometimes called Briggsian or 'vulgar logarithms , but the term which ultimately took hold was common logarithms. Nearly any unstated base of a logarithm, except in certain applications, was assumed to be 10; with the practical use of logarithm tables now rare, this is no longer the case.

Natural logarithms

Main article: natural logarithm.

There is a special base e (approximately 2.718281828459045) which has useful properties. The logarithm to this base is called the natural logarithm. When dealing with the logarithms to the base $e$, it is common especially to denote $\log _{e}$ by $\ln$ especially if there is any likelihood that the reader might think that base 10 or base 2 logarithms might be meant. In most pure mathematical work, $\log$ or $\ln$ is used to denote $\log _{e}$; in most engineering work, $\log$ means $\log _{10}$; while in information theory, $\log$ often means $\log _{2}$, which is traditionally written as $\mathrm {ld}$ (from the Latin logarithmus dualis), but also sometimes is written as $\lg$. Whenever a possibility for ambiguity exists, this ambiguity is resolved by explicitly writing out the base.

Historically, base e logarithms were sometimes called Naperian logarithms, slightly misleadingly since Napier's system did not have log(1)=0; the term which ultimately took hold was natural logarithms.

Binary logarithms

As mentioned, the base used extensively in information theory and computer science is the binary logarithm, base 2. It is used frequently because many algorithms and computer applications split items into two sub-items, in the divide-and-conquer manner. Binary logarithms are useful in determining characteristics of the time or space complexity of such algorithms. The base of the logarithm is hardly ever mentioned explicitly when analyzing the asymptotic complexity of algorithms in terms of Big O notation, since $O(\log _{b}n)=O(\log _{c}n)$ for all valid bases b and c. Shannon's law relies on the use of base-2 logarithms when considering the basic unit of information the bit, which is generally the most useful modality.

Change of base

One's choice of base with logarithms is not crucial, because a logarithm can be converted from one base to another quite easily. For example, to calculate the value of a logarithm of a base other than 10, given a table or calculator that can only handle base 10, the following formula changes the base to any chosen base (assuming that a, b, and k are all positive real numbers and that $a\neq 1$ and $k\neq 1$)

$\log _{a}b={\frac {\log _{k}b}{\log _{k}a}}$

where k is any valid base. Letting k=b gives

$\log _{a}b={\frac {1}{\log _{b}a}}.$

In particular we have:

$\log _{e}10=2.30258509$

$\log _{10}e=0.434294482$

$\log _{2}10=3.32192809$

$\log _{10}2=0.301029996$

$\log _{2}e=1.44269504$

Tables of logarithms

Prior to the advent of computers and calculators, using logarithms meant using tables of logarithms, which had to be created manually. Base-10 logarithms are useful in computations when electronic means are not available. See common logarithm for details, including the use of characteristics and mantissas of common (i.e., base-10) logarithms.

In 1617, Briggs published the first installment of his own table of common logarithms, containing the logarithms of all integers below 1000 to eight places of decimals. This he followed, in 1624, by his Arithmetica Logarithmica, containing the logarithms of all integers from 1 to 20,000 and from 90,000 to 100,000 to fourteen places of decimals, together with a learned introduction, in which the theory and use of logarithms are fully developed. The interval from 20,000 to 90,000 was filled up by Adrian Vlacq, a Dutch computer; but in his table, which appeared in 1628, the logarithms were given to only ten places of decimals.

Vlacq's table was later to 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." (Athenaeum, 15 June 1872. See also the Monthly Notices of the Royal Astronomical Society for May 1872.) An edition of Vlacq's work, containing many corrections, was issued at Leipzig in 1794 under the title Thesaurus Logarithmorum Completus by George Vega.

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 Mr. Sang 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.

Besides the tables mentioned above, a great collection, called Tables du Cadastre, was constructed under the direction of Prony, by an original computation, under the auspices of the French republican government of the 1700's. 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." (English Cyclopaedia, Biography, Vol. IV., article "Prony.") Cubic interpolation could be used to find the logarithm of any number to a similar accuracy.

To the modern student who has the benefit of a calculator, the work put into the tables just mentioned is a small indication of the importance of logarithms.

Applications in calculus

To calculate the derivative of a logarithmic function, the following formula is used

${\frac {d}{dx}}\log _{b}x={\frac {1}{x\ln b}}={\frac {\log _{b}e}{x}}$

where ln is the natural logarithm, i.e. with base e. Letting b=e:

${\frac {d}{dx}}\ln(x)={\frac {1}{x}},\qquad \int {\frac {1}{x}}\,dx=\ln x+C$

One can then see that the following formula gives the integral of a logarithm

$\int \log _{b}(x)\,dx=x\log _{b}(x)-{\frac {x}{\ln b}}+C=x\log _{b}\left({\frac {x}{e}}\right)+C$

Logarithms of complex numbers

Logarithms may also be defined for complex arguments. This is explained on the natural logarithm page.

Group theory

In the theory of finite groups there is a related notion of discrete logarithm. For some finite groups, it is believed that the discrete logarithm is very hard to calculate, whereas discrete exponentials are quite easy. This asymmetry has applications in cryptography.

Relationship between binary and common logarithms

A curious coincidence is the approximation log₂(x) ≈ log₁₀(x) + ln(x), accurate to about 99.4% or 2 significant digits; this is because ¹/_ln 2 − ¹/_ln 10 is approximately 1 (in fact 1.0084...). Another interesting coincidence is that, approximately, log₁₀2 = 0.3 (the actual value is about 0.301029995); this corresponds to the fact that, with an error of only 2.4%, 2¹⁰ ≈ 10³ (i.e. 1024 is about 1000; see also Binary prefix).

Source

Much of the history of logarithms is derived from The Elements of Logarithms with an Explanation of the Three and Four Place Tables of Logarithmic and Trigonometric Functions, by James Mills Peirce, University Professor of Mathematics in Harvard University, 1873.

See also

• Natural logarithm
• Logarithmic identities