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In mathematics, a 'logarithm' (to base ''b'') of a number ''x'' is written $log\_b(x)$ or, if unambiguous, $log(x)$, and equals the exponent ''y'' that satisfies ''x'' = ''b''^''y''.
In other words
: $y=log\_b(x),!$
is equivalent to
: $x=\; b^y,!$
The base ''b'' must be neither 0 nor 1 (nor a root of 1 in the case of the extension to complex numbers, the complex logarithm), and is typically 10, ''e'', or 2. When ''x'' and ''b'' are further restricted to positive real numbers, the logarithm is a unique real number.
For example, since
:$3^4\; =\; 3\; imes\; 3\; imes\; 3\; imes\; 3\; =\; 81,\; ,$
:$log\_3(81)\; =\; 4,\; ,$
or, in words, the base-3 logarithm of 81 is 4, or the log base-3 of 81 is 4.

Contents

The logarithm as a function

Integer and non-integer exponents

Bases

Other notations

Change of base

Uses of logarithms

Science and engineering

Exponential functions

Easier computations

Calculus

Series for calculating the natural logarithm

Computers

Generalizations

History

Tables of logarithms

See also

References

External links

The logarithm as a function

The function log_''b''(''x'') depends on both ''b'' and ''x'', but the term 'logarithm function' (or 'logarithmic function') in standard usage refers to a function of the form log_''b''(''x'') in which the 'base' ''b'' is fixed and so the only argument is ''x''. Thus there is one logarithm function for each value of the base ''b'' (which must be positive and must differ from 1).
Viewed in this way, the base-b logarithm function is the inverse function of the exponential function ''b''^''x''. The word "logarithm" is often used to refer to a logarithm function itself as well as to particular values of this function.

Integer and non-integer exponents

If ''n'' is a positive integer, ''b''^''n'' signifies the product of ''n'' factors equal to ''b'':
:$underbrace\{b\; imes\; b\; imes\; cdots\; imes\; b\}\_n.$
However, if ''b'' is a positive real number not equal to 1, this definition can be extended to any real number ''n'' in a field (see exponentiation). Similarly, the logarithm function can be defined for any positive real number. For each positive base ''b'' not equal to 1, there is one logarithm function and one exponential function, which are inverses of each other.
Logarithms can reduce multiplication operations to addition, division to subtraction, exponentiation to multiplication, and roots to division. Therefore, logarithms are useful for making lengthy numerical operations easier to perform and, before the advent of electronic computers, they were widely used for this purpose in fields such as astronomy, engineering, navigation, and cartography. They have important mathematical properties and are still widely used today.

Bases

The most widely used bases for logarithms are 10, the mathematical constant ''e'' ≈ 2.71828... and 2. When "log" is written without a base (''b'' missing from log_''b''), the intent can usually be determined from context:

★ natural logarithm (log_''e'', ln, log, or Ln) in mathematical analysis, statistics, economics and some engineering fields

★ common logarithm (log₁₀ or simply log; sometimes lg) in some engineering fields and when logarithm tables are used to simplify hand calculations

★ binary logarithm (log₂; sometimes lg or lb) in information theory and musical intervals

★ indefinite logarithm when the base is irrelevant, e.g. in complexity theory when describing the asymptotic behavior of algorithms in big O notation.
To avoid confusion, it is best to specify the base if there is any chance of misinterpretation.

Other notations

The notation "ln(''x'')" invariably means log_e(''x''), i.e., the natural logarithm of ''x'', but the implied base for "log(''x'')" varies by discipline:

★ Mathematicians generally understand both "ln(''x'')" and "log(''x'')" to mean log_e(''x'') and write "log₁₀(''x'')" when the base-10 logarithm of ''x'' is intended. Calculus textbooks will occasionally write "lg(''x'')" to represent "log₁₀(''x'')".

★ Many engineers, biologists, astronomers, and some others write only "ln(''x'')" or "log_e(''x'')" when they mean the natural logarithm of ''x'', and take "log(''x'')" to mean log₁₀(''x'') or, sometimes in the context of computing, log₂(''x'').

★ On most calculators, the LOG button is log₁₀(''x'') and LN is log_e(''x'').

★ In most commonly used computer programming languages, including C, C++, Java, Fortran, Ruby, and BASIC, the "log" function returns the natural logarithm. The base-10 function, if it is available, is generally "log10."

★ Some people use Log(''x'') (capital ''L'') to mean log₁₀(''x''), and 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.

★ A notation frequently used in some European countries is the notation ^''b''log(''x'') instead of log_''b''(''x'').^[1]
This chaos, historically, originates from the fact that the natural logarithm has nice mathematical properties (such as its derivative being 1/''x'', and having a simple definition), while the base 10 logarithms, or decimal logarithms, were more convenient for speeding calculations (back when they were used for that purpose). Thus natural logarithms were only extensively used in fields like calculus while decimal logarithms were widely used elsewhere.
As recently as 1984, Paul Halmos in his "automathography" ''I Want to Be a Mathematician'' 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, many mathematicians have adopted the "ln" notation, but most use "log".
In computer science, the base 2 logarithm is sometimes written as lg(''x''), as suggested by Edward Reingold and popularized by Donald Knuth. However, lg(''x'') is also sometimes used for the common log, and lb(''x'') for the binary log. Mathematics: from the birth of numbers., Gullberg, Jan, , , W. W. Norton & Co, 1997, In Russian literature, the notation lg(''x'') is also generally used for the base 10 logarithm.^[2]
In German, lg(''x'') also denotes the base 10 logarithm, while sometimes ld(''x'') or lb(''x'') is used for the base 2 logarithm.
The clear advice of the United States Department of Commerce National Institute of Standards and Technology is to follow the ISO standard ''Mathematical signs and symbols for use in physical sciences and technology, ISO 31-11:1992'', which suggests these notations:^[3]

★ The notation "ln(''x'')" means log_e(''x'');

★ The notation "lg(''x'')" means log₁₀(''x'');

★ The notation "lb(''x'')" means log₂(''x'').

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 ''k'':
:
Moreover, this result implies that all logarithm functions (whatever the base) are similar to each other. So to calculate the log with base 2 of the number 16 with a calculator:
:

Uses of logarithms

Logarithms are useful in solving equations in which exponents are unknown. They have simple derivatives, so they are often used in the solution of integrals. The logarithm is one of three closely related functions. In the equation ''b''^''n'' = ''x'', ''b'' can be determined with radicals, ''n'' with logarithms, and ''x'' with exponentials. See logarithmic identities for several rules governing the logarithm functions.

Science and engineering

Various quantities in science are expressed as logarithms of other quantities; see logarithmic scale for an explanation and a more complete list.

★ In chemistry, the negative of the base-10 logarithm of the concentration of hydronium ions (H₃O⁺, the form H⁺ takes in water) is the measure known as pH. The concentration of hydronium ions in neutral water is 10⁻⁷ mol/L at 25 °C, hence a pH of 7.

★ The ''bel'' (symbol B) is a unit of measure which is the base-10 logarithm of ratios, such as power levels and voltage levels. It is mostly used in telecommunication, electronics, and acoustics. It is used, in part, because the ear responds logarithmically to acoustic power. The Bel is named after telecommunications pioneer Alexander Graham Bell. The ''decibel'' (dB), equal to 0.1 bel, is more commonly used. The ''neper'' is a similar unit which uses the natural logarithm of a ratio.

★ The Richter scale measures earthquake intensity on a base-10 logarithmic scale.

★ In spectrometry and optics, the absorbance unit used to measure optical density is equivalent to −1 B.

★ In astronomy, the apparent magnitude measures the brightness of stars logarithmically, since the eye also responds logarithmically to brightness.

★ In psychophysics, the Weber–Fechner law proposes a logarithmic relationship between stimulus and sensation.

★ In computer science, logarithms often appear in bounds for computational complexity. For example, to sort ''N'' items using comparison can require time proportional to ''N'' log ''N''.

★ Many types of engineering and scientific data are typically graphed on log-log or semilog axes, in order to most clearly show the form of the data.

Exponential functions

The exponential function $e^x,$ also written as $mathrm\{exp\}(x),$ is defined as the inverse of the natural logarithm. It is positive for every real argument $x.$
The operation of "raising $b$ to a power $p$" for positive arguments $b$ and all real exponents $p$ is defined by
:$b^p\; =\; exp(\{pln\; b\; \}).,$
The 'antilogarithm' function is another name for the inverse of the logarithmic function. It is written antilog_''b''(''n'') and means the same as $b^n$.

Easier computations

Logarithms switch the focus from normal numbers to exponents. As long as the same base is used, this makes certain operations easier; in this table, upper-case variables represent logs of corresponding lower-case variables:
{| style="border:1px solid" align="center"
|- align="center"
! Operation with numbers !! Operation with exponents !! Logarithmic identity
|- align="center"
| $!,\; a\; b$ || $!,\; A\; +\; B$ || $!,\; log(a\; b)\; =\; log(a)\; +\; log(b)$
|- align="center"
| $!,\; a\; /\; b$ || $!,\; A\; -\; B$ || $!,\; log(a\; /\; b)\; =\; log(a)\; -\; log(b)$
|- align="center"
| $!,\; a\; ^\; b$ || $!,\; Ab$ || $!,\; log(a\; ^\; b)\; =\; b\; log(a)$
|- align="center"
| $!,\; sqrt[b]\{a\}$ || $!,\; A\; /\; b$ ||
|}
These relations typically made such operations on two numbers much faster and the proper use of logarithms was an essential skill before multiplying calculators became available.
The $log(a\; b)\; =\; log(a)\; +log(b)$ equation is fundamental (it implies effectively the other three relations in a field) because it
describes an isomorphism between the 'additive group' and the 'multiplicative group' of the field.
To multiply two numbers, one found the logarithms of both numbers on a table of common logarithms, added them, and then looked up the result in the table to find the product. This is faster than multiplying them by hand, provided that more than two decimal figures are needed in the result. The table needed to get an accuracy of seven decimals could be fit in a big book, and the table for nine decimals occupied a few shelves.
The discovery of logarithms just before Newton's era had an impact in the scientific world which can be compared with the invention of the computer in the 20th century, because many calculations which were too laborious became feasible.
When the chronometer was invented in the 18th century, logarithms allowed all calculations needed for astronomical navigation to be reduced to just additions, speeding the process by one or two orders of magnitude. A table of logarithms with five decimals, plus logarithms of trigonometric functions, was enough for most astronomical navigation calculations, and those tables fit in a small book.
To compute powers or roots of a number, the common logarithm of that number was looked up and multiplied or divided by the radix. Interpolation could be used for still higher precision. Slide rules used logarithms to perform the same operations more rapidly, but with much less 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.

Calculus

The derivative of the natural logarithm function is
:
By applying the change-of-base rule, the derivative for other bases is
:
The antiderivative of the natural logarithm ln(''x'') is
: $int\; ln(x)\; ,dx\; =\; x\; ln(x)\; -\; x\; +\; C,$
and so the antiderivative of the logarithm for other bases is
:
ight) + C.
''See also:'' , list of integrals of logarithmic functions.

Series for calculating the natural logarithm

There are several series for calculating natural logarithms.^[4] The simplest, though inefficient, is:
: when
To derive this series, start with
:
Integrate both sides to obtain
:
:
Letting $z\; =\; 1-x\; !$ and thus $x\; =\; -(z-1)\; !$, we get
:
A more efficient series is
:
ight ) }^{2n+1}
for ''z'' with positive real part.
To derive this series, we begin by substituting −''x'' for ''x'' and get
:
Subtracting, we get
:
Letting and thus , we get
:
ight)}^3 + rac{1}{5}{left(rac{z-1}{z+1}
ight)}^5 + cdots
ight ).
For example, applying this series to
:
we get
:
and thus
:
rac{1}{7 cdot 1000000} + cdots
ight )
:$=\; 0.2\; cdot\; (1.0000000dots\; +\; 0.0033333dots\; +\; 0.0000200dots\; +\; 0.0000001dots\; +\; cdots)$
:$=\; 0.2\; cdot\; 1.0033535dots\; =\; 0.2006707dots$
where we factored 1/10 out of the sum in the first line.
For any other base ''b'', we use
:

Computers

Most computer languages use log(''x'') for the natural logarithm, while the common log is typically denoted log10(x). The argument and return values are typically a floating point (or double precision) data type.
As the argument is floating point, it can be useful to consider the following:
A floating point value ''x'' is represented by a mantissa ''m'' and exponent ''n'' to form
: $x\; =\; m2^n.,$
Therefore
: $ln(x)\; =\; ln(m)\; +\; nln(2).,$
Thus, instead of computing $ln(x)$ we compute $ln(m)$ for some ''m'' such that 1 ≤ ''m'' ≤  2. Having ''m'' in this range means that the value is always in the range . Some machines uses the mantissa in the range and in that case the value for u will be in the range In either case, the series is even easier to compute.
The integer part of the logarithm to base 2 of an integer is given by the position of the left-most bit, and can be very quickly computed using the following algorithm:
int log2(int x){
int r = 0;
while( (x >> r) != 0){
r++;
}
return r;
}
This algorithm can execute quickly using very few processor instructions.

Generalizations

The ordinary logarithm of positive reals generalizes to negative and complex arguments, though it is a multivalued function that needs a branch cut terminating at the branch point at 0 to make an ordinary function or principal branch. The logarithm (to base ''e'') of a complex number ''z'' is the complex number ln(|''z''|) + ''i'' arg(''z''), where |''z''| is the modulus of ''z'', arg(''z'') is the argument, and ''i'' is the imaginary unit; see complex logarithm for details.
The discrete logarithm is a related notion in the theory of finite groups. It involves solving the equation ''b''^''n'' = ''x'', where ''b'' and ''x'' are elements of the group, and ''n'' is an integer specifying a power in the group operation. 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 public key cryptography.
The logarithm of a matrix is the inverse of the matrix exponential.
A 'double logarithm', $ln(ln(x))$, 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''.
For each positive ''b'' not equal to 1, the function log_''b''  (''x'') is an isomorphism from the group of positive real numbers under multiplication to the group of (all) real numbers under addition. They are the only such isomorphisms that are continuous. The logarithm function can be extended to a Haar measure in the topological group of positive real numbers under multiplication.

History

The method of logarithms was first publicly propounded in 1614, in a book entitled ''Mirifici Logarithmorum Canonis Descriptio,'' by John Napier,^[5]
Baron of Merchiston in Scotland (Joost Bürgi independently discovered logarithms; however, he did not publish his discovery until four years after Napier). 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 used constantly in surveying, navigation, and other branches of practical mathematics. It supplanted the more involved method of 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 higher theoretical mathematics.
At first, Napier called logarithms "artificial numbers" and antilogarithms "natural numbers". Later, Napier formed the word ''logarithm'' to mean a number that indicates a ratio: (''logos'') meaning proportion, and (''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 (Bürgi chose ''r'' = 1 + 10⁻⁴ = 1.0001). Napier's original logarithms did not have log 1 = 0 but rather log 10⁷ = 0. Thus if ''N'' is a number and ''L'' is its logarithm as calculated by Napier, ''N'' = 10⁷(1 − 10⁻⁷)^''L''. Since (1 − 10⁻⁷)¹⁰⁷ is approximately 1/''e'', this makes ''L''/10⁷ approximately equal to log_1/''e'' ''N''/10⁷.

Tables of logarithms

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

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, Henry 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 Adriaan Vlacq, a Dutch mathematician; but in his table, which appeared in 1628, the logarithms were given to only ten places of decimals.
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."^[6] 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 Mr. 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.
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 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." ^[7] Cubic interpolation could be used to find the logarithm of any number to a similar accuracy.

See also

★ List of logarithm topics

★ Logarithmic identities

★ Logarithmic scale

★ Natural logarithm

★ Common logarithm

★ Complex logarithm

★ Imaginary-base logarithm

★ Indefinite logarithm

★ Iterated logarithm

★ Logarithmic units

★ Discrete logarithm

★ Zech's logarithms

★ Logarithm of a matrix

★ Log-normal distribution

★ Decibel (expressing loudness on a logarithmic scale)

★ Equal temperament classifying pitch on a logarithmic scale

★ Richter scale

★ pH

References

1. "Mathematisches Lexikon" at Mateh_online.at
2.
"Common Logarithm" at MathWorld
3. Guide for the Use of the International System of Units (SI) B. N. Taylor
4. ''Handbook of Mathematical Functions'', National Bureau of Standards (Applied Mathematics Series no.55), June 1964, page 68.
5. 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.
6. ''Athenaeum,'' 15 June 1872. See also the ''Monthly Notices of the Royal Astronomical Society'' for May 1872.
7. ''English Cyclopaedia, Biography,'' Vol. IV., article "Prony."

External links

★ Explaining Logarithms

★ Log Calculator for all bases.

★ Logarithm on MathWorld

★ Jost Burgi, Swiss Inventor of Logarithms

★ Logarithm calculators and word problems with work shown, for school students

★ Translation of Napier's work on logarithms

★ Logarithms - from The Little Handbook of Statistical Practice

★ Algorithm for determining Log values for any base