Dictionary Definition
logarithm n : the exponent required to produce a
given number [syn: log]
User Contributed Dictionary
English
Etymology
From logarithmus, term coined by Scot mathematician John Napier from λόγος (logos) "word, reason" and αριθμός (arithmos) "number".Noun
Translations
 Croatian: logaritam
 Czech: logaritmus
 Dutch: logaritme
 Finnish: logaritmi
 French: logarithme
 German: Logarithmus
 Hungarian: logaritmus
 Italian: logaritmo
 Polish: logarytm
 Portuguese: logaritmo
 Spanish: logaritmo
 Swedish: logaritm
Synonyms
Derived terms
See also
Extensive Definition
In mathematics, the logarithm
of a given number to a given base
is the power
or exponent to which
the base must be raised in order to produce the given number.
For example, the logarithm of 1000 to the common
base 10 is 3, because 10 raised to the power of 3 is 1000; the base
2 logarithm of 32 is 5 because 2 to the power 5 is 32.
The logarithm of x to the base b is written
logb(x) or, if the base is implicit, as log(x). So, for a number x,
a base b and an exponent y,
 \mbox~~ b^y = x, ~~\mbox~~ \log_b (x) = y \,.
An important feature of logarithms is that they
reduce multiplication to addition, by the formula:
 \log (x \times y) = \log x + \log y \,.
That is, the logarithm of the product of two
numbers is the sum of the logarithms of those numbers. The use of
logarithms to facilitate complex calculations was a significant
motivation in their original development.
Properties of the logarithm
main article List of logarithmic identities When x and b are restricted to positive real numbers, logb(x) is a unique real number. The magnitude of the base b must be neither 0 nor 1; the base used is typically 10, e, or 2. Logarithms are defined for real numbers and for complex numbers.The major property of logarithms is that they map
multiplication to addition. This ability stems from the following
identity:
 b^x \times b^y = b^ \ ,
which by taking logarithms becomes
 \log_b \left(b^x \times b^y \right) = \log_b \left( b^ \right) \ = x + y = \log_b \left(b^x \right) + \log_b \left(b^y \right). \
A related property is reduction of exponentiation
to multiplication. Using the identity:
 c = b^ \ ,
 c^p = \left(b^\right)^p = b^ \ ,
 \log_b \left(c^p \right) = p \log_b (c ) \ .
In words, to raise a number to a power p, find
the logarithm of the number and multiply it by p. The exponentiated
value is then the inverse logarithm of this product; that is,
number to power = bproduct.
Besides reducing multiplication operations to
addition, and exponentiation to multiplication, logarithms reduce
division to subtraction, and roots to division. Logarithms make
lengthy numerical operations easier
to perform. The whole process is made easy by using tables
of logarithms, or a slide rule,
antiquated now that calculators are available. Although the above
practical advantages are not important for numerical work today,
they are used in graphical analysis (see Bode
plot).
The logarithm as a function
Though logarithms have been traditionally thought
of as arithmetic sequences of numbers corresponding to geometric
sequences of other (positive real) numbers, as in the 1797
Britannica definition, they are also the result of applying an
analytic
function. The function can therefore be meaningfully extended
to complex numbers.
The function logb(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 logb(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 baseb
logarithm function is the inverse
function of the exponential function bx.
The word "logarithm" is often used to refer to a logarithm function
itself as well as to particular values of this function.
The base can also be a complex number; the
evaluation of the log is just slightly more complicated in this
case. See
imaginary base.
Logarithm of a complex number
When the base b is real and z is a complex
number, say z = x + i y, the
logarithm of z is found easily by putting z in polar
form that is, z =
(x2 + y2)1/2 exp (i tan−1
(y / x) ). If the base of the logarithm is chosen
as e , that is, using loge (denoted by ln and called the natural
logarithm), the logarithm becomes:
 \ln(z) = \ln \left[ \left( x^2 + y^2 \right) ^ e^\right]


 = \ln \left[ \left( x^2 + y^2 \right) ^\right] + \ln \left[ e^\right]
 =\frac \ln \left( x^2 + y^2 \right) + i \tan^\left( \frac \right) \ .

This evaluation uses the properties of all
logarithms (see above), regardless of choice of base:
logb (c d ) = logb (c
) + logb (d ) and its generalization to
arbitrary products logb bz = z. Because the inverse
tangent is a multiple valued function of its argument, the
logarithm of a complex number is not unique either. See article on
complex logarithm.
Group theory
From the pure mathematical perspective, the
identity
 \log(cd) = \log(c) + \log(d) \,
is fundamental in two senses. First, the
remaining three arithmetic properties can be derived from it.
Furthermore, it expresses an isomorphism between the
multiplicative
group of the positive real numbers and the additive
group of all the reals.
Logarithmic functions are the only continuous
isomorphisms from the multiplicative group of positive real numbers
to the additive group of real numbers.
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 logb), the intent can usually be determined from
context:
 natural logarithm (loge, ln, log, or Ln) in mathematical analysis, statistics, economics and some engineering fields. The reasons to consider e the natural base for logarithms, though perhaps not obvious, are numerous and compelling.
 common logarithm (log10 or simply log; sometimes lg) in various engineering fields, especially for power levels and power ratios, such as acoustical sound pressure, and in logarithm tables to be used to simplify hand calculations
 binary logarithm (log2; sometimes lg, lb, or ld) in information theory and computerrelated fields
 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 loge(x),
i.e., the natural
logarithm of x, but the implied base for "log(x)" varies by
discipline:
 Mathematicians generally understand "ln(x)" to mean loge(x) and "log(x)" to mean "log10(x)". Calculus textbooks will occasionally write "lg(x)" to represent "log10(x)".
 On most calculators, the LOG button is log10(x) and LN is loge(x).
 Some people use Log(x) (capital L) to mean log10(x), and use log(x) with a lowercase l to mean loge(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 blog(x) instead of logb(x).
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. In Russian literature, the
notation lg(x) is also generally used for the base 10 logarithm. 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 3111:1992, which suggests these
notations:
 The notation "ln(x)" means loge(x);
 The notation "lg(x)" means log10(x);
 The notation "lb(x)" means log2(x).
As the difference between logarithms to different
bases is one of scale, it is possible to consider all logarithm
functions to be the same, merely giving the answer in different
units, such as dB, neper, bits, decades, etc.; see the section
Science
and engineering below. Logarithms to a base less than 1 have a
negative scale, or a flip about the x axis, relative to logarithms
of base greater than 1.
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 loge and
log10). To find a logarithm with base b, using any other base
k:
 \log_b(x) = \frac.
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:
 \log_2(16) = \frac.
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 bn = 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
Various quantities in science are expressed as
logarithms of other quantities; see logarithmic
scale for an explanation and a more complete list.
 The bel (symbol B) is a unit of measure which is the base10 logarithm of ratios, such as power levels and voltage levels. It is mostly used in telecommunication, electronics, and acoustics. 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 base10 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 the product N × log N. Similarly, base2 logarithms are used to express the amount of storage space or memory required for a binary representation of a number—with k bits (each a 0 or a 1) one can represent 2k distinct values, so any natural number N can be represented in no more than (log2 N) + 1 bits.
 Similarly, in information theory logarithms are used as a measure of quantity of information. If a message recipient may expect any one of N possible messages with equal likelihood, then the amount of information conveyed by any one such message is quantified as log2 N bits.
 In geometry the logarithm is used to form the metric for the halfplane model of hyperbolic geometry.
 In inferential statistics, the logarithm of the data in a dataset can be used for parametric statistical testing if the original data does not meet the assumption of normality.
 Musical intervals are measured logarithmically as semitones. The interval between two notes in semitones is the base21/12 logarithm of the frequency ratio (or equivalently, 12 times the base2 logarithm). Fractional semitones are used for nonequal temperaments. Especially to measure deviations from the equal tempered scale, intervals are also expressed in cents (hundredths of an equallytempered semitone). The interval between two notes in cents is the base21/1200 logarithm of the frequency ratio (or 1200 times the base2 logarithm). In MIDI, notes are numbered on the semitone scale (logarithmic absolute nominal pitch with middle C at 60). For microtuning to other tuning systems, a logarithmic scale is defined filling in the ranges between the semitones of the equal tempered scale in a compatible way. This scale corresponds to the note numbers for whole semitones. (see microtuning in MIDI).
Exponential functions
One way of defining the exponential
function ex, also written as exp(x), is 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 = \left( e^ \right) ^p = e^.\,
The antilogarithm function is another name for
the inverse of the logarithmic function. It is written antilogb(n)
and means the same as bn.
Easier computations
Logarithms can be used to replace difficult
operations on numbers by easier operations on their logs (in any
base), as the following table summarizes. In the table, uppercase
variables represent logs of corresponding lowercase variables:
These arithmetic properties of logarithms make such calculations
much faster. The use of logarithms was an essential skill until
electronic computers
and calculators
became available. Indeed the discovery of logarithms, just before
Newton's era, had an impact in the scientific world that can be
compared with that of the advent of computers in the 20th century
because it made feasible many calculations that had previously been
too laborious.
As an example, to approximate the product of two
numbers one can look up their logarithms in a table,
add them, and, using the table again, proceed from that sum to its
antilogarithm, which is the desired product. The precision of the
approximation can be increased by interpolating between
table entries. For manual calculations that demand any appreciable
precision, this process, requiring three lookups and a sum, is much
faster than performing the multiplication. To achieve seven decimal
places of accuracy requires a table that fills a single large
volume; a table for ninedecimal accuracy occupies a few shelves.
Similarly, to approximate a power cd one can look up log c in the
table, look up the log of that, and add to it the log of d; roots
can be approximated in much the same way.
One key application of these techniques was
celestial navigation. Once the invention of the chronometer
made possible the accurate measurement of longitude at sea, mariners had
everything necessary to reduce their navigational computations to
mere additions. A fivedigit table of logarithms and a table of the
logarithms of trigonometric
functions sufficed for most purposes, and those tables could
fit in a small book. Another critical application with even broader
impact was the slide rule, an
essential calculating tool for engineers. Many of the powerful
capabilities of the slide rule derive from a clever but simple
design that relies on the arithmetic properties of logarithms. The
slide rule allows computation much faster still than the techniques
based on tables, but provides much less precision, although slide
rule operations can be chained to calculate answers to any
arbitrary precision.
Related operations
Cologarithms
The cologarithm of a number is the logarithm of
the inverse of said number, meaning cologb(x)=logb(1/x)= 
logb(x).
Antilogarithms
The antilogarithm is the logarithmic inverse of
the logarithm, meaning that the antilogb(logb(x))=x. Thus, setting
by=x implies that logb(x)=y. By taking the antilogb of both sides,
antilogb(logb(x))=antilogby, thus x=antilogby. Therefore,
by=antilogby.
Calculus
The natural logarithm of a positive number x can
be defined as
 \ln (x) \equiv \int_^ \frac.
The derivative of the natural
logarithm function is
 \frac \ln(x) = \frac.
By applying the changeofbase rule, the
derivative for other bases is
 \frac \log_b(x) = \frac \frac = \frac = \frac.
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
 \int \log_b(x) \,dx = x \log_b(x)  \frac + C = x \log_b \left(\frac\right) + C.
Series for calculating the natural logarithm
There are several series for calculating natural
logarithms. The simplest, though inefficient, is:
 \ln (z) = \sum_^\infty \frac (z1)^n when z1
To derive this series, start with (x)
 \frac = 1 + x + x^2 + x^3 + \cdots.
Integrate both sides to obtain
 \ln(1x) = x + \frac + \frac + \cdots
 \ln(1x) = x  \frac  \frac  \frac  \cdots.
Letting z = 1x \! and thus x = (z1) \!, we get
 \ln z = (z1)  \frac + \frac  \frac + \cdots
A more efficient series is
 \ln (z) = 2 \sum_^\infty \frac ^
for z with positive real part.
To derive this series, we begin by substituting
−x for x and get
 \ln(1+x) = x  \frac + \frac  \frac + \cdots.
Subtracting, we get
 \ln \frac = \ln(1+x)  \ln(1x) = 2x + 2\frac + 2\frac + \cdots.
Letting z = \frac \! and thus x = \frac \!, we
get
 \ln z = 2 \left ( \frac + \frac^3 + \frac^5 + \cdots \right ).
For example, applying this series to
 z = \frac,
we get
 \frac = \frac = \frac,
and thus
 \ln (1.2222222\dots) = \frac \left (1 + \frac + \frac +
 = 0.2 \cdot (1.0000000\dots + 0.0033333\dots + 0.0000200\dots + 0.0000001\dots + \cdots)
 = 0.2 \cdot 1.0033535\dots = 0.2006707\dots
where we factored 1/10 out of the sum in the
first line.
For any other base b, we use
 \log_b (x) = \frac.
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) + n\ln(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 u = \frac is always in
the range 0 \le u . Some machines use the mantissa in the range 0.5
\le m and in that case the value for u will be in the range
\frac13 In either case, the series is even easier to
compute.
To compute a base 2 logarithm on a number between
1 and 2 in an alternate way, square it repeatedly. Every time it
goes over 2, divide it by 2 and write a "1" bit, else just write a
"0" bit. This is because squaring doubles the logarithm of a
number.
The integer part of the logarithm to base 2 of an
unsigned integer is given by the position of the leftmost bit, and
can be computed in O(n) steps using the following algorithm:
int log2(int x)
However, it can also be computed in O(log n)
steps by trying to shift by powers of 2 and checking that the
result stays nonzero: for example, first >>16, then
>>8, ... (Each step reveals one bit of the result)
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 bn = 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.
It is possible to take the logarithm of a
quaternions and
octonions.
A double logarithm, \ln(\ln(x)), is the inverse
function of the
double exponential function. A superlogarithm
or hyper4logarithm
is the inverse function of tetration. The superlogarithm
of x grows even more slowly than the double logarithm for large
x.
For each positive b not equal to 1, the function
logb (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,
Baron of Merchiston, in Scotland.
(Joost
Bürgi independently discovered logarithms; however, he did not
publish his discovery until four years after Napier.) Early
resistance to the use of logarithms was muted by Kepler's
enthusiastic support and his publication of a clear and impeccable
explanation of how they worked.
Their use contributed to the advance of science,
and especially of astronomy, by making some difficult calculations
possible. Prior to the advent of calculators and computers, they
were 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 the
utility of the logarithm concept in computation, the natural
logarithm presented a solution to the problem of quadrature
of a hyperbolic
sector at the hand of Gregoire
de SaintVincent in 1647.
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 they represent, 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
log1/e N/107. 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 sevenplace table (Paris, 1795), instead of
stopping at 100,000, gave the eightplace 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 sevenplace tables. The only
important published extension of Vlacq's table was made by Mr. Sang
in 1871, whose
table contained the sevenplace 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
twentyfour 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.
See also
 List of logarithm topics
 List of logarithmic identities
 Logarithmic scale
 Natural logarithm
 Common logarithm
 Complex logarithm
 Imaginarybase logarithm
 Indefinite logarithm
 Iterated logarithm
 Logarithmic units
 Discrete logarithm
 Zech's logarithms
 Logarithm of a matrix
 Lognormal distribution
 Decibel
 Equal temperament
 Richter magnitude scale
 pH
References
External links
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