Logarithm

In mathematics, a logarithm is a function that gives the exponent in the equation bⁿ = x. It is usually written as log_b x = n. For example:

\!\,3^{4}=81{\mbox{, thus }}\log _{3}81=4

The logarithm is one of three closely related functions. With bⁿ = x, b can be determined with radicals, n with logarithms, and x with exponentials.

The negative of a logarithm is written as n = −log_b x; an example of its use is in chemistry, where it expresses the concentration of protons (pH).

An antilogarithm is used to show the inverse of the logarithm. It is written antilog_b(n) and means the same as bⁿ.

A double logarithm is the inverse function of the double-exponential function. A super-logarithm or hyper-logarithm is the inverse function of the super-exponential function. The super-logarithm of x grows even more slowly than the double logarithm for large x.

A discrete logarithm is a related notion in the theory of finite groups. 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.

Logarithms and exponentials: inverses

For each base (b in bⁿ), there is one logarithm function and one exponential function; they are inverse functions. With bⁿ = x:

Exponentials determine x when given n; to find x, they multiply 1 by b as many times as (n).
Logarithms determine n when given x; n is the number of times that x must be divided by b to reach 1.

Using logarithms

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

For integers b and x, the number log_b(x) is irrational (i.e., not a quotient of two integers) if one of b and x has a prime factor which the other does not (and in particular if they are coprime and both greater than 1). In certain cases this fact can be proved very quickly: for example, if log₂3 were rational, we would have log₂3 = n/m for some positive integers n and m, thus implying 2ⁿ = 3^m. But this last identity is impossible, since 2ⁿ is even and 3^m is odd.

Unspecified bases

Mathematicians generally understand either "ln(x)" or "log(x)" to mean log_e(x), i.e., the natural logarithm of x, and write "log₁₀(x)" if the base-10 logarithm of x is intended.

Engineers, biologists, and some others write only "ln(x)" or (occasionally) "log_e(x)" when they mean the natural logarithm of x, and take "log(x)" to mean log₁₀(x) or, in the context of computing, log₂(x).

Sometimes Log(x) (capital L) is used to mean log₁₀(x), by those people who use log(x) with a lowercase l to mean log_e(x).

The notation Log(x) is also used by mathematicians to denote the principal branch of the (natural) logarithm function.

In most commonly used programming languages, including C, C++, Fortran, and BASIC, "log" or "LOG" means natural logarithm.

Most of the reason for thinking about base-10 logarithms became obsolete shortly after about 1970 when hand-held calculators became widespread (for more on this point, see common logarithm). Nonetheless, since calculators are made and often used by engineers, the conventions to which engineers were accustomed continued to be used on calculators, so now most non-mathematicians take "log(x)" to mean the base-10 logarithm of x and use only "ln(x)" to refer to the natural logarithm of x. As recently as 1984, Paul Halmos in his autobiography heaped contempt on what he considered the childish "ln" notation, which he said no mathematician had ever used. (The notation was in fact invented in 1893 by Irving Stringham, professor of mathematics at Berkeley.) As of 2005, some mathematicians have adopted the "ln" notation, but most use "log". In computer science, the base 2 logarithm is written as lg(x) to avoid confusion. This usage was suggested by Edward Reingold and popularized by Donald Knuth.

When "log" is written without a base (b missing from log_b), the intent can usually be determined from context:

natural logarithm (log_e) in mathematical analysis;
binary logarithm (log₂) with musical intervals and in subjects that deal with bits;
common logarithm (log₁₀) when logarithm tables are used to simplify hand calculations;
indefinite logarithm when the base is irrelevant.

Change of base

While there are several useful identities, the most important for calculator use lets one find logarithms with bases other than those built into the calculator (usually log_e and log₁₀). To find a logarithm with base b using any other base a:

\log _{b}(x)={\frac {\log _{a}(x)}{\log _{a}(b)}}

**Base change formula proof**
$b^{\log _{b}(x)}=x\!\,$	by definition
$\log _{k}\left(b^{\log _{b}(x)}\right)=\log _{k}(x)$	take logs on both sides
$\log _{b}(x)\,\log _{k}(b)=\log _{k}(x)$	simplify the left hand side
$\log _{b}(x)={\frac {\log _{k}(x)}{\log _{k}(b)}}\,\!$	divide by log_k(b)

All this implies, moreover, that all logarithm functions (whatever the base) are similar to each other.

Uses of logarithms

Logarithms are useful in solving equations in which exponents are unknown. They have simple derivatives, so they are often used as the solution of integrals. Furthermore, various quantities in science are expressed as logarithms of other quantities; see logarithmic scale for an explanation and a list.

Exponential functions

Sometimes (especially in the context of analysis) it is necessary to calculate arbitrary exponential functions $f(x)^{x}$ using only the natural exponent $e^{x}$ :

$f(x)^{x}=e^{\log(f(x)^{x})}$
$=e^{x\log(f(x))}$

Easier computations

Logarithms switch the focus from normal numbers to exponents. As long as the same base is used, this makes a few operations easier:

Operation with numbers	Operation with exponents	Logarithmic identity
$\!\,ab$	$\!\,A+B$	$\!\,\log(ab)=\log(a)+\log(b)$
$\!\,a/b$	$\!\,A-B$	$\!\,\log(a/b)=\log(a)-\log(b)$
$\!\,a^{b}$	$\!\,Ab$	$\!\,\log(a^{b})=b\log(a)$
$\!\,{\sqrt[{b}]{a}}$	$\!\,A/b$	$\!\,\log({\sqrt[{b}]{a}})=\log(a)/b$

Before electronic calculators, this made difficult operations on two numbers much easier. One simply found the logarithms of both numbers (multiply and divide) or the first number (power or root, where one number is already an exponent) in a table of common logarithms, performed a simpler operation on those, and found the result on a table. Slide rules performed the same operations using logarithms, but faster and with lower precision than using tables. Other tools for performing multiplications before the invention of the calculator include Napier's bones and mechanical calculators (see history of computing hardware).

In abstract algebra, this property of the logarithm functions can be summarized by noting that any logarithm function with a fixed base is a group isomorphism from the group of strictly positive real numbers under multiplication to the group of all real numbers under addition.

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

History

Joost Bürgi, a Swiss clockmaker in the employ of the Duke of Hesse-Kassel, first conceived of logarithms. The method of natural logarithms was first propounded in 1614, in a book entitled Mirifici Logarithmorum Canonis Descriptio, by John Napier (c. 1550 - 1618; Latinized Neperus), Baron of Merchiston in Scotland, four years after the publication of his memorable invention. This method contributed to the advance of science, and especially of astronomy, by making some difficult calculations possible. Prior to the advent of calculators and computers, it was constantly used in surveying, navigation, and other branches of practical mathematics. It supplanted the more involved prosthaphaeresis, which relied on trigonometric identities, as a quick method of computing products. Besides their usefulness in computation, logarithms also fill an important place in the higher theoretical mathematics.

At first, Napier called logarithms "artificial numbers" and antilogarithms "natural numbers". Later, Napier formed the word logarithm, a portmanteau, to mean a number that indicates a ratio: λoγoς (logos) meaning ratio, and αριθμoς (arithmos) meaning number. Napier chose that because the difference of two logarithms determines the ratio of the numbers for which they stand, so that an arithmetic series of logarithms corresponds to a geometric series of numbers. The term antilogarithm was introduced in the late 17th century and, while never used extensively in mathematics, persisted in collections of tables until they fell into disuse.

Napier did not use a base as we now understand it, but his logarithms were, up to a scaling factor, effectively to base $1/e$ . For interpolation purposes and ease of calculation, it is useful to make the ratio $r$ in the geometric series close to 1. Napier chose $r=1-10^{-7}=0.999999$ , and Bürgi chose $r=1+10^{-4}=1.0001$ . Napier's original logarithms did not have log 1 = 0 but rather log $10^{7}$ = 0. Thus if $N$ is a number and $L$ is its logarithm as calculated by Napier, $N=10^{7}(1-10^{-7})^{L}$ . Since $(1-10^{-7})$ is approximately $1/e$ , $L$ is approximately $10^{7}\log _{1/e}N/10^{7}$ .

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 decimal places. 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 Jurij 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 1700s. 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.

Algorithm

To calculate log_b(x) if b and x are rational numbers and x ≥ b > 1:

If n₀ is the largest natural number such that b^n₀ ≤ x or, alternately,

n_{0}=\lfloor \log _{b}(x)\rfloor

then

\log _{b}(x)=n_{0}+{\frac {1}{\log _{x/b^{n_{0}}}(b)}}

This algorithm recursively produces the continued fraction

\log _{b}(x)=n_{0}+{\frac {1}{n_{1}+{\frac {1}{n_{2}+{\frac {1}{n_{3}+\cdots }}}}}}.

The logarithms produced are irrational for most inputs.

To use irrational numbers as inputs, apply the algorithm to successively detailed rational approximations. The limit of the result series should converge to the actual result.

\ln 2=2\sum _{n=0}^{\infty }{\frac {(1/5)^{2\,n+1}}{2\,n+1}}+2\sum _{n=0}^{\infty }{\frac {(1/7)^{2\,n+1}}{2\,n+1}}

**Algorithm proof**
$\log _{b}(x)=\log _{b}(x)\,\!$	identity
$\log _{b}(x)=n_{0}+\log _{b}(x)-n_{0}\,\!$	algebraic manipulation
$\log _{b}(x)=n_{0}+\log _{b}(x)-\log _{b}(b^{n_{0}})\,\!$	logarithmic identity
$\log _{b}(x)=n_{0}+\log _{b}\left({\begin{matrix}{\frac {x}{b^{n_{0}}}}\end{matrix}}\right)$	logarithmic identity
$\log _{b}(x)=n_{0}+{\frac {1}{\log _{\begin{matrix}{\frac {x}{b^{n_{0}}}}\end{matrix}}(b)}}$	base switch

Trivia

Unicode glyph

log has its own Unicode glyph: ㏒ (U+33D2 or 13266 in decimal). This is more likely due to its presence in Asian legacy encodings than its importance as a mathematical function.

Alternate notation

A few people use the notation ^blog(x) instead of log_b(x).

Relationships between binary, natural and common logarithms

In particular we have:

log₂(e) ≈ 1.44269504

log₂(10) ≈ 3.32192809

log_e(10) ≈ 2.30258509

log_e(2) ≈ 0.693147181

log₁₀(2) ≈ 0.301029996

log₁₀(e) ≈ 0.434294482

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) ≈ 1 (in fact 1.0084...). The property is demonstrated in all six conversion factors above, arranged in pairs of two:

2.30	3.32
0.30	0.69
0.43	1.44

This comes on top of the reciprocal relations we have:

2.30	0.43
0.30	3.32
0.69	1.44

Another interesting coincidence is that log₁₀(2) ≈ 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).

References

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.

External links