Mathematics is the discipline that deals with concepts such as quantity, structure, space and change.... , the logarithm of a number to a given baseBase (mathematics)
In arithmetic, the base refers to the number b in an expression of the form b'n.... is the power or exponent to which the base must be raised in order to produce the number.
For example, the logarithm of 1000 to the 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,
An important feature of logarithms is that they reduce multiplication to addition, by the formula:
That is, the logarithm of the product of two numbers is the sum of the logarithms of those numbers.
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1614John Napier publishes a paper outlining his discovery of logarithms.
Mathematics is the discipline that deals with concepts such as quantity, structure, space and change.... , the logarithm of a number to a given baseBase (mathematics)
In arithmetic, the base refers to the number b in an expression of the form b'n.... is the power or exponent to which the base must be raised in order to produce the number.
For example, the logarithm of 1000 to the 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,
An important feature of logarithms is that they reduce multiplication to addition, by the formula:
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
When x and b are restricted to positive real numberReal number
In mathematics, the set of real numbers, denoted R, is the set of all rational numbers and irrational numbers.... s, logb(x) is a unique real number. The magnitudeAbsolute value
In mathematics, the absolute value of a real number is its numerical value without regard to its sign.... of the base b must be neither 0 nor 1; the base used is typically 10, eE (mathematical constant)
The mathematical constant e is the base of the natural logarithm.... , 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:
which by taking logarithms becomes
A related property is reduction of exponentiation to multiplication. Using the identity:
it follows that c to the power p (exponentiation) is: or, taking logarithms:
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 performLogarithm
The logarithm is the mathematical operation that is the inverse of exponentiation .... . The whole process is made easy by using tables of logarithmsLogarithm
The logarithm is the mathematical operation that is the inverse of exponentiation .... , or a slide ruleSlide rule
The slide rule is a mechanical analog computer, consisting of at least two finely divided scales , most often a fixed outer... , 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 plotBode plot
A Bode plot, named for Hendrik Wade Bode, is usually a combination of a Bode magnitude plot and Bode phase 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 functionAnalytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series.... . 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 baseBase (mathematics)
In arithmetic, the base refers to the number b in an expression of the form b'n.... 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 functionInverse function
In mathematics, an inverse function is in simple terms a function which "does the reverse" of a given function.... of the exponential functionExponentiation
Exponentiation is a mathematical operation, written a'n, involving two numbers, the base a and the ... 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 baseComplex logarithm
One may define the natural logarithm also for all non-zero complex numbers z, but it is usually denoted log for mostly two rea... .
Logarithm of a complex number
When the base b is real and z is a complex numberComplex number
In mathematics, a complex number is a number of the form ... , say z = x + iy, the logarithm of z is found 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 logarithmLogarithm
The logarithm is the mathematical operation that is the inverse of exponentiation .... ), the logarithm becomes:
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 logbbz = 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 logarithmComplex logarithm
One may define the natural logarithm also for all non-zero complex numbers z, but it is usually denoted log for mostly two rea... .
Group theory
From the pure mathematical perspective, the identity
is fundamental in two senses. First, the remaining three arithmetic properties can be derived from it. Furthermore, it expresses an isomorphismFacts About Isomorphism
In mathematics, an isomorphism is a bijective map f such that both f and its inverse f −1 are homomo... between the multiplicative groupMultiplicative group
In mathematics, multiplicative group in group theory may mean... of the positive real numbers and the additive groupAdditive group Summary
In mathematics, an additive group may be... 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 eE (mathematical constant)
The mathematical constant e is the base of the natural logarithm.... ˜ 2.71828... and 2. When "log" is written without a base (b missing from logb), the intent can usually be determined from context:
The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is equal ... (logeE (mathematical constant)
The mathematical constant e is the base of the natural logarithm.... , ln, log, or Ln) in mathematical analysisMathematical analysis
Analysis is a branch of mathematics that depends upon the concepts of limits and convergence.... , statisticsStatistics
Statistics is a mathematical science pertaining to the collection, analysis, interpretation, and presentation of data.... , economicsEconomics Overview
In the social sciences, economics is the study of the production, distribution, and consumption of goods and services..... and some engineeringEngineering
Engineering is the application of scientific and mathematical principles to develop economical solutions to technical proble... fields. The reasons to consider e the natural base for logarithms, though perhaps not obvious, are numerous and compelling.
In mathematics, the common logarithm is the logarithm with base 10.... (log10 or simply log; sometimes lg) in various engineeringEngineering
Engineering is the application of scientific and mathematical principles to develop economical solutions to technical proble... fields, especially for power levels and power ratios, such as acoustical sound pressureSound pressure
Sound pressure is the pressure deviation from the local ambient pressure caused by a sound wave.... , and in logarithm tableMathematical table
Before calculators were cheap and plentiful, people would use mathematical tables —lists of numbers showing the result... s to be used to simplify hand calculations
In mathematics, the binary logarithm is the logarithm for base 2.... (log2; sometimes lg, lb, or ld), in computer scienceComputer science Summary
Computer science, or computing science, is the study of the theoretical foundations of information and computation and... and information theoryFacts About Information theory
Information theory is a discipline in applied mathematics involving the quantification of data with the goal of enabling as ...
The indefinite logarithm of a positive number or even sometimes just ) is just the logarithm with respect to any base, when... when the base is irrelevant, e.g. in complexity theoryComputational complexity theory
In computer science, computational complexity theory is the branch of the theory of computation that studies the resources, ... when describing the asymptotic behavior of algorithmAlgorithm
In mathematics and computing, an algorithm is a procedure for accomplishing some task which, given an initial state, will t... s in big O notationBig O notation
Big O notation or Big Oh notation, and also Landau notation or asymptotic notation, is a mathematical nota... .
To avoid confusion, it is best to specify the base if there is any chance of misinterpretation.
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:
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 derivativeDerivative
In mathematics, the derivative is defined as the instantaneous rate of change of a function.... s, so they are often used in the solution of integralIntegral
In calculus, the integral of a function is an extension of the concept of a sum.... s. 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 exponentialsExponential function
The exponential function is one of the most important functions in mathematics.... . See logarithmic identities for several rules governing the logarithm functions.
The exponential function is one of the most important functions in mathematics.... 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
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, upper-case variables represent logs of corresponding lower-case variables:
These arithmetic properties of logarithms make such calculations much faster. The use of logarithms was an essential skill until electronic computerComputer
A computer is a machine for manipulating data according to a list of instructions known as a program.... s and calculatorFacts About Calculator
A calculator is a device for performing calculations.... s 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 interpolatingInterpolation
In the mathematical subfield of numerical analysis, interpolation is a method of constructing new data points from a discret... 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 nine-decimal 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 chronometerMarine chronometer
A marine chronometer is a timekeeper precise enough to be used as a portable time standard, used to determine longitude by m... made possible the accurate measurement of longitudeLongitude
Longitude, sometimes denoted by the Greek letter ? , describes the location of a place on Earth east or west of a north-sout... at sea, mariners had everything necessary to reduce their navigational computations to mere additions. A five-digit 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 ruleSlide rule
The slide rule is a mechanical analog computer, consisting of at least two finely divided scales , most often a fixed outer... , 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
The following is a list of integrals of logarithmic functions.... .
Series for calculating the natural logarithm
There are several series for calculating natural logarithms. The simplest, though inefficient, is: when
To derive this series, start with ()
Integrate both sides to obtain
Letting and thus , we get
A more efficient series is
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
For example, applying this series to
we get
and thus
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 pointFloating point
Floating-point is a means of representing real numbers in terms of digits or bits in a computer or calculator, similar to ho... (or double precisionDouble precision
In computing, double precision is a computer numbering format that occupies two storage locations in computer memory at addr... ) data type.
Floating-point is a means of representing real numbers in terms of digits or bits in a computer or calculator, similar to ho... , it can be useful to consider the following:
A floating point value x is represented by a mantissaSignificand
The significand is the part of a floating-point number that contains its significant digits.... m and exponent n to form
Therefore
Thus, instead of computing we compute for some m such that 1 ≤ m < 2. Having m in this range means that the value is always in the range . Some machines use 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.
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 left-most 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 complexComplex number
In mathematics, a complex number is a number of the form ... arguments, though it is a multivalued functionMultivalued function
In mathematics, a multivalued function is a total relation; i.e.... that needs a branch cut terminating at the branch pointBranch point
In complex analysis, a branch point may be thought of informally as a point z0 at which a "multiple-valued function" cha... at 0 to make an ordinary function or principal branchPrincipal branch
In mathematics, a principal branch is a function which selects one branch, or "slice", of a multi-valued function.... . The logarithm (to base e) of a complex number z is the complex number ln(|z|) + i arg(z), where |z| is the modulusAbsolute value
In mathematics, the absolute value of a real number is its numerical value without regard to its sign.... of z, arg(z) is the argumentComplex number
In mathematics, a complex number is a number of the form ... , and i is the imaginary unitImaginary unit
In mathematics, the imaginary unit allows the real number system to be extended to the complex number system .... ; see complex logarithmComplex logarithm
One may define the natural logarithm also for all non-zero complex numbers z, but it is usually denoted log for mostly two rea... for details.
In mathematics, specifically in abstract algebra and its applications, discrete logarithms are group-theoretic analogues of ... is a related notion in the theory of finite groupFinite group
In mathematics, a finite group is a group which has finitely many elements.... s. 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.
In mathematics, the logarithm of a matrix is a generalization of the scalar logarithm to matrices.... is the inverse of the matrix exponentialMatrix exponential
In mathematics, the matrix exponential is a function on square matrices analogous to the ordinary exponential function.... .
It is possible to take the logarithm of a quaternionQuaternion
In mathematics, quaternions are a non-commutative extension of complex numbers.... s and octonionOctonion
In mathematics, the octonions are a nonassociative extension of the quaternions.... s.
A double exponential function is a constant raised to the power of an exponential function.... . A super-logarithmSuper-logarithm
In mathematics, the super-logarithm is one of the two inverse functions of tetration.... or hyperFacts About Hyper operator
The hyper operators forming the hypern family are as follows:... -4-logarithm is the inverse function of tetrationTetration
Tetration is iterated exponentiation, the first hyper operator after exponentiation.... . The super-logarithmSuper-logarithm
In mathematics, the super-logarithm is one of the two inverse functions of tetration.... 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 isomorphismIsomorphism Overview
In mathematics, an isomorphism is a bijective map f such that both f and its inverse f −1 are homomo... from the groupGroup (mathematics)
In mathematics, a group is a set together with a binary operation satisfying certain axioms, detailed below.... 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 measureHaar measure Overview
In mathematical analysis, the Haar measure is a way to assign an "invariant volume" to subsets of locally compact topologica... in the topological groupTopological group
In mathematics, a topological groupG is a group that is also a topological space such that the group multiplication G... 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 NapierJohn Napier
John Napier or Neper, nicknamed Marvellous Merchiston was a Scottish mathematician, physicist, astronomer/astro... , Baron of Merchiston, in ScotlandScotland Summary
Scotland is a nation in northwest Europe and one of the constituent countries of the United Kingdom.... . Early resistance to the use of logarithms was muted by Kepler'sJohannes Kepler
Johannes Kepler , a key figure in the scientific revolution, was a German mathematician, astronomer, astrologer, and an earl... 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 prosthaphaeresisProsthaphaeresis
Prosthaphaeresis was an algorithm used in the late 16th century and early 17th century for approximating products using form... , 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 quadratureNumerical integration
In numerical analysis, numerical integration constitutes a broad family of algorithms for calculating the numerical value of... of a hyperbolic sectorHyperbolic sector
A hyperbolic sector is a region of the Cartesian plane bounded by rays from the origin to two points and and by the hyper... at the hand of Gregoire de Saint-VincentGrégoire de Saint-Vincent
Gr?goire de Saint-Vincent a Jesuit, was a mathematician who discovered that the area under a rectangular hyperbola is the ... 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: (logosLogos
The Greek word ????? or logos is a word with various meanings.... ) 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 seriesGeometric series
In mathematics, a geometric series is a series with a constant ratio between successive terms.... 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/eN/107.
A computer is a machine for manipulating data according to a list of instructions known as a program.... s and calculatorCalculator
A calculator is a device for performing calculations.... s, 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 logarithmCommon logarithm
In mathematics, the common logarithm is the logarithm with base 10.... for details, including the use of characteristics and mantissaMantissa
Mantissa can mean:* The "fraction part" of the result of a logarithm.... s of common (i.e., base-10) logarithms.
Henry Briggs was an English mathematician notable for changing Napier's logarithms into common/Briggesian logarithms.... published the first installment of his own table of common logarithms, containing the logarithms of all integers below 1000 to eight decimalDecimal
The decimal numeral system has ten as its base.... 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 VlacqAdriaan Vlacq
Adriaan Vlacq was a Dutch book publisher and mathematician.... , 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." An edition of Vlacq's work, containing many corrections, was issued at LeipzigLeipzig Summary
Leipzig [] is the largest city in the federal state of Saxony in Germany with a population of 502,000.... in 1794 under the title Thesaurus Logarithmorum Completus by Jurij VegaFacts About Jurij Vega
Baron Jurij Bartolomej Vega was a Slovenian mathematician, physicist and artillery officer.... .
François Callet's seven-place table, 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 interpolationInterpolation
In the mathematical subfield of numerical analysis, interpolation is a method of constructing new data points from a discret... , 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 functionTrigonometric function
In mathematics, the trigonometric functions are functions of an angle; they are important when studying triangles and modeli... s.
Besides the tables mentioned above, a great collection, called Tables du Cadastre, was constructed under the direction of Gaspard de PronyGaspard de Prony
Gaspard Clair Franois Marie Riche de Prony was a French mathematician and engineer, who worked on hydraulics.... , by an original computation, under the auspices of the FrenchFrance
France, officially the French Republic, is a country whose metropolitan territory is located in Western Europe and whi... republican government of the 1700s1700s
Events and trends*The Bonneville Slide blocks the Columbia River near the site of present-day Cascade Locks, Oregon with a land b... . 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." CubicCubic function Overview
In mathematics, a cubic function is a function of the form... interpolationInterpolation
In the mathematical subfield of numerical analysis, interpolation is a method of constructing new data points from a discret... could be used to find the logarithm of any number to a similar accuracy.
