Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere.... , the logarithm of a number to a given base
Base (mathematics)
In arithmetic, the base refers to the number b in an expression of the form bn. The number n is called the exponent and the expression is known formally as exponentiation of b by n or the exponential of n with base b.... 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 3 is how many 10s you must multiply to get 1000: thus 10 × 10 × 10 = 1000; the base 2 logarithm of 32 is 5 because 5 is how many 2s one must multiply to get 32: thus 2 × 2 × 2 × 2 × 2 = 32.
Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere.... , the logarithm of a number to a given base
Base (mathematics)
In arithmetic, the base refers to the number b in an expression of the form bn. The number n is called the exponent and the expression is known formally as exponentiation of b by n or the exponential of n with base b.... 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 3 is how many 10s you must multiply to get 1000: thus 10 × 10 × 10 = 1000; the base 2 logarithm of 32 is 5 because 5 is how many 2s one must multiply to get 32: thus 2 × 2 × 2 × 2 × 2 = 32. In the language of exponents: 103 = 1000, so log101000 = 3, and 25 = 32, so log232 = 5.
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 number
Real number
In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co... s, logb(x) is a unique real number. The magnitude
Absolute value
In mathematics, the absolute value of a real number is its numerical value without regard to its Negative and non-negative numbers. So, for example, 3 is the absolute value of both 3 and -3.... of the base b must be neither 0 nor 1; the base used is typically 10, e
E (mathematical constant)
The mathematical constant e is the unique real number such that the function ex has the same value as the derivative, for all values of x.... , 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
For example,
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.
For example,
Besides reducing multiplication operations to addition, and exponentiation to multiplication, logarithms reduce division to subtraction, and roots to division. For example,
In mathematics, the logarithm of a number to a given base is the Power or exponent to which the base must be raised in order to produce the number.... . The whole process is made easy by using tables of logarithms
Logarithm
In mathematics, the logarithm of a number to a given base is the Power or exponent to which the base must be raised in order to produce the number.... , or a slide rule
Slide rule
The slide rule, also known colloquially as a slipstick, is a mechanical analog computer. The slide rule is used primarily for multiplication and division , and also for "scientific" functions such as Nth roots, logarithms and trigonometry, but does not generally perform addition or subtraction.... , 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
Bode plot
A Bode magnitude plot is a plot of logarithm magnitude versus frequency, plotted with a log-frequency axis, to show the transfer function or frequency response of a LTI system theory system.... ).
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
Analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. Analytic functions can be thought of as a bridge between polynomials and general functions.... . 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
Base (mathematics)
In arithmetic, the base refers to the number b in an expression of the form bn. The number n is called the exponent and the expression is known formally as exponentiation of b by n or the exponential of n with base b.... 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
Inverse function
In mathematics, if ƒ is a function from A to B then an inverse function for ƒ is a function in the opposite direction, from B to A, with the property that a round trip from A to B to A returns each element of the initial set to itself.... of the exponential function
Exponentiation
Exponentiation is a mathematics operation , written 'an', involving two numbers, the base a and the exponentn.... bx. The word "logarithm" is often used to refer to a logarithm function itself as well as to particular values of this function.
Logarithm of a negative or complex number
There is no real-valued logarithm for negative or non-real complex number
Complex number
In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:... s. The logarithm function can be extended to the complex logarithm
Complex logarithm
In complex analysis, a complex logarithm function is an "inverse function" of the complex exponential function, just as the natural logarithm ln x is the inverse of the exponential function ex.... , which does apply to these cases. The value is not unique though, since for example which implies that both and 0 are equally valid logarithms to base e of 1.
When is a complex number, say , the logarithm of is found by putting in polar form that is, where and is any angle such that and . The function arg
Arg (mathematics)
In mathematics, arg is a function operating on complex numbers , and intuitively gives the angle between the line joining the point to the origin and the positive real number Cartesian coordinate system, shown as in figure 1 opposite, known as an argument of the point .... is a multi-valued function.
If the base of the logarithm is chosen as e , that is, using (denoted by and called the natural logarithm
Logarithm
In mathematics, the logarithm of a number to a given base is the Power or exponent to which the base must be raised in order to produce the number.... ), the complex logarithm
Complex logarithm
In complex analysis, a complex logarithm function is an "inverse function" of the complex exponential function, just as the natural logarithm ln x is the inverse of the exponential function ex.... is:
which is, just like arg, also a multi-valued function. The principal value of the logarithm, Log (denoted by a capital first letter), is a single-valued function and is defined as
In mathematics, arg is a function operating on complex numbers , and intuitively gives the angle between the line joining the point to the origin and the positive real number Cartesian coordinate system, shown as in figure 1 opposite, known as an argument of the point .... is the principal argument. It is a single-valued function and defined as the branch
Branch point
In the mathematics field of complex analysis, a branch point of a multivalued function is a point such that the function is discontinuous when going around an arbitrarily small circuit around this point .... of in which the values are in the range leaving a branch cut at the negative reals. The principal argument of any positive real number is 0; hence the principal logarithm of such a number is always real and equals the natural logarithm.
The principal value of the logarithm of a negative number is:
For a base b other than e the complex logarithm can be defined as , the principal value of which is given by the principal values of and .
Exponentiation is a mathematics operation , written 'an', involving two numbers, the base a and the exponentn.... .
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 isomorphism
Isomorphism
In abstract algebra, an isomorphism is a bijection map f such that both f and its inverse function f −1 are homomorphisms, i.e., structure-preserving mappings.... between
the multiplicative group
Multiplicative group
In mathematics and group theory the term multiplicative group refers to one of the following concepts, depending on the context*any group whose binary operation is written in multiplicative notation ,... of the positive real numbers and the additive group
Additive group
In mathematics, an additive group may be*an abelian group, when it is written using the symbol + for its binary operation*the underlying group under addition of a field , ring , vector space or other structure having addition as one of its operations... 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
E (mathematical constant)
The mathematical constant e is the unique real number such that the function ex has the same value as the derivative, for all values of x.... ˜ 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 an irrational number constant approximately equal to 2.718281828.... (loge
E (mathematical constant)
The mathematical constant e is the unique real number such that the function ex has the same value as the derivative, for all values of x.... , ln, log, or Ln) in mathematical analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of calculus. It is the branch of mathematics most explicitly concerned with the notion of a limit , whether the limit of a sequence or the limit of a function.... , statistics
Statistics
Statistics is a Mathematics pertaining to the collection, analysis, interpretation or explanation, and presentation of data. It also provides tools for prediction and forecasting based on data.... , economics
Economics
File:Ballard Farmers' Market - vegetables.jpgEconomics is the Social sciences that studies the Production theory basics, Distribution , and Consumption of Good and Service .... and some engineering
Engineering
Engineering is the discipline and profession of applying Technology and science knowledge and utilizing natural laws and physical resources in order to design and implement materials, structures, machines, devices, systems, and process that safely realize a desired objective and meet specified criteria.... fields. The reasons to consider e the natural base for logarithms, though perhaps not obvious, are numerous and compelling.
The common logarithm is the logarithm with base 10. It is also known as the decadic logarithm, named after its base. It is indicated by log10, or sometimes Log with a capital L .... (log10 or simply log; sometimes lg) in various engineering
Engineering
Engineering is the discipline and profession of applying Technology and science knowledge and utilizing natural laws and physical resources in order to design and implement materials, structures, machines, devices, systems, and process that safely realize a desired objective and meet specified criteria.... fields, especially for power levels and power ratios, such as acoustical sound pressure
Sound pressure
Sound pressure is the local pressure deviation from the ambient pressure caused by a sound wave. Sound pressure can be measured using a microphone in air and a hydrophone in water.... , and in logarithm table
Mathematical table
Before calculators were cheap and plentiful, people would use mathematical tables —lists of numbers showing the results of calculation with varying arguments— to simplify and drastically speed up computation.... s to be used to simplify hand calculations
In mathematics, the binary logarithm is the logarithm for base 2. It is the inverse function of .... (log2; sometimes lg, lb, or ld), in computer science
Computer science
Computer science is the study of the theoretical foundations of information and computation, and of practical techniques for their implementation and application in computer systems.... and information theory
Information theory
Information theory is a branch of applied mathematics and electrical engineering involving the quantification of information. Historically, information theory was developed by Claude E....
The indefinite logarithm of a positive number is the logarithm without regard to any particular base: it is a function , not a number.... (Log or [log ] or simply log) when the base is irrelevant, e.g. in complexity theory
Computational complexity theory
Computational complexity theory, as a branch of the theory of computation in computer science, investigates the problems related to the Computational resource required for the execution of algorithms , and the inherent difficulty in providing efficient algorithms for specific computational problems.... when describing the asymptotic behavior of algorithm
Algorithm
In mathematics, computing, linguistics and related subjects, an algorithm is a sequence of finite instructions, often used for calculation and data processing.... s in big O notation
Big O notation
In mathematics, big O notation describes the asymptotic analysis of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions.... .
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
Natural logarithm
The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e , where e is an irrational number constant approximately equal to 2.718281828.... of x, but the implied base for "log(x)" varies by discipline:
Mathematicians understand "log(x)" to mean loge(x). Calculus textbooks will occasionally write "lg(x)" to represent "log10(x)".
Many engineers, biologists, astronomers, and some others write only "ln(x)" or "loge(x)" when they mean the natural logarithm of x, and take "log(x)" to mean log10(x) or, in computer science
Computer science
Computer science is the study of the theoretical foundations of information and computation, and of practical techniques for their implementation and application in computer systems.... , log2
Binary logarithm
In mathematics, the binary logarithm is the logarithm for base 2. It is the inverse function of .... (x).
On most calculators, the LOG button is log10(x) and LN is loge(x).
A programming language is a machine-readable artificial language designed to express computations that can be performed by a machine, particularly a computer.... s, including C
C (programming language)
C is a general-purpose computer programming language originally developed in 1972 by Dennis Ritchie at the Bell Telephone Laboratories to implement the Unix operating system.... , C++
C++
C++ is a general-purpose programming language. It is regarded as a middle-level language, as it comprises a combination of both high-level programming language and low-level programming language language features.... , Java
Java (programming language)
Java is a programming language originally developed by James Gosling at Sun Microsystems and released in 1995 as a core component of Sun Microsystems' Java .... , Haskell
Haskell (programming language)
Haskell is a standardized, purely functional programming language with non-strict programming language, named after logician Haskell Curry. The goals of the language are described as:... , Fortran
Fortran
Fortran is a general-purpose programming language, procedural programming language, imperative programming language programming language that is especially suited to numerical analysis and scientific computing.... , Python
Python (programming language)
Python is a general-purpose high-level programming language. Its design philosophy emphasizes code readability. Python's core syntax and semantics are Minimalism , while the standard library is large and comprehensive.... , Ruby
Ruby (programming language)
Ruby is a dynamic programming language, reflection , general purpose object-oriented programming language that combines syntax inspired by Perl with Smalltalk-like features.... , 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 loge(x).
The notation Log(x) is also used by mathematicians to denote the principal branch
Principal branch
In mathematics, a principal branch is a function which selects one branch, or "slice", of a multi-valued function. Most often, this applies to functions defined on the complex plane: see branch cut.... of the (natural) logarithm function.
In some European countries, a frequently used notation is 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
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point.... 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.
Paul Richard Halmos was a Hungary-born Jewish United States mathematician who made fundamental advances in the areas of probability theory, statistics, operator theory, ergodic theory, functional analysis , and mathematical logic.... 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
University of California, Berkeley
The University of California, Berkeley is a public university research university located in Berkeley, California, California, United States. The oldest of the ten major campuses affiliated with the University of California, Berkeley offers some 300 undergraduate and graduate degree programs in a wide range of disciplines.... .
In computer science, the base 2 logarithm is sometimes written as lg(x), as suggested by Edward Reingold
Edward Reingold
Edward M. Reingold is a computer scientist active in the fields of algorithms, data structures and calendrical calculations.He has co-authored the standard text on calendrical calculations, Calendrical Calculations, with Nachum Dershowitz.... and popularized by Donald Knuth
Donald Knuth
Donald Ervin Knuth is a renowned computer science and Emeritus of the Art of Computer Programming at Stanford University.Author of the seminal multi-volume work The Art of Computer Programming , Knuth has been called the "father" of the run-time analysis, contributing to the development of, and systematizing formal mathematical techn... . 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 United States Department of Commerce is the United States Cabinet department of the United States Federal government of the United States concerned with promoting economic growth.... National Institute of Standards and Technology
National Institute of Standards and Technology
The National Institute of Standards and Technology , known between 1901 and 1988 as the National Bureau of Standards , is a measurement standards laboratory which is a non-regulatory agency of the United States Department of Commerce.... is to follow the ISO
International Organization for Standardization
The International Organization for Standardization , widely known as ISO , is an international standard-setting body composed of representatives from various national standards organizations.... standard Mathematical signs and symbols for use in physical sciences and technology, ISO 31-11:1992, 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:
This is because the definition of logarithm says that
but we can also get a by using the base k logarithm and then get
with b ? 1, because logk 1 = 0. Any number to the power of 0 is equal to 1.
Moreover, this result implies 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 derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point.... s, so they are often used in the solution of integral
Integral
Integration is an important concept in mathematics, specifically in the field of calculus and, more broadly, mathematical analysis. Given a function ƒ of a Real number variable x and an interval [a, b] of the real line, the integral... s. The logarithm is one of three closely related functions. In the equation bn = x, b can be determined with radical
Nth root
In mathematics, an nth root of a number a is a number b such that when n copies of b are multiplication together, the result is a.... s, n with logarithms, and x with exponentials
Exponential function
The exponential function is a function in mathematics. The application of this function to a value x is written as exp. Equivalently, this can be written in the form ex, where e is the mathematical constant that is the base of the natural logarithm and that is also known as Euler's number.... . 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
Logarithmic scale
A logarithmic scale is a scale that uses the logarithm of a physical quantity instead of the quantity itself.Presentation of data on a logarithmic scale can be helpful when the data covers a large range of values – the logarithm reduces this to a more manageable range.... for an explanation and a more complete list.
Chemistry is the science concerned with the composition, structure, and properties of matter, as well as the changes it undergoes during chemical reactions.... , the negative of the base-10 logarithm of the activity
Activity (chemistry)
In chemical thermodynamics activity is a measure of the ?effective concentration? of a species in a mixture. By convention, it is a dimensionless quantity.... of hydronium
Hydronium
In chemistry, hydronium is the common name for the aqueous cation hydrogen3oxygen+ derived from protonation of water. It is the simplest type of an oxonium ion.... ions (H3O+, the form H+ takes in water) is the measure known as pH
PH
pH is a measure of the Acid or Base of a solution. It is defined as the cologarithm of the Activity of dissolved hydrogen ions . Hydrogen ion activity coefficients cannot be measured experimentally, so they are based on theoretical calculations.... . The activity of hydronium ions in neutral water
Water
Water is a common chemical substance that is essential for the survival of all known forms of life. In typical usage, water refers only to its liquid form or States of matter, but the substance also has a solid state, ice, and a gaseous state, water vapor or steam.... is 10−7 mol/L at 25 °C, hence a pH of 7.
The definition, agreement and practical use of units of measurement have played a crucial role in human endeavour from early ages up to this day.... of measure which is the base-10 logarithm of ratio
Ratio
A ratio is an expression which compares quantities relative to each other. The most common examples involve two quantities, but in theory any number of quantities can be compared.... s, such as power
Power (physics)
In physics, power is the rate at which mechanical work is performed or energy is transmitted, or the amount of energy required or expended for a given unit of time.... levels and voltage
Voltage
Electrical tension is the potential difference between two points of an electrical or electronic circuit, expressed in volts. It is the measurement of the potential for an electric field to cause an electric current in an electrical conductor.... levels. It is mostly used in telecommunication
Telecommunication
Telecommunication is the assisted Transmission of Signal over a distance for the purpose of communication. In earlier times, this may have involved the use of smoke signals, Drum , Semaphore line, flag signals or heliograph.... , electronics
Electronics
Electronics refers to the flow of charge through nonmetal electrical conductor , whereas electrical refers to the flow of charge through metal electrical conductor.... , and acoustics
Acoustics
Acoustics is the interdisciplinary science that deals with the study of sound, ultrasound and infrasound . A scientist who works in the field of acoustics is an acoustician.... . The Bel is named after telecommunications pioneer Alexander Graham Bell
Alexander Graham Bell
Alexander Graham Bell was an eminent scientist, Innovation and innovator who is credited with inventing the first practical telephone.Bell's father, grandfather, and brother had all been associated with work on elocution and speech, and both his mother and wife were deaf, profoundly influencing Bell's life's work.... . The decibel
Decibel
The decibel is a logarithmic units of measurement that expresses the magnitude of a physical quantity relative to a specified or implied reference level.... (dB), equal to 0.1 bel, is more commonly used. The neper
Neper
A neper is a logarithmic unit of ratio. It is not an SI unit but is accepted for use alongside the SI. It is used to express ratios, such as gain and loss, and relative values.... is a similar unit which uses the natural logarithm of a ratio.
The Richter magnitude scale, or more correctly local magnitudeML scale, assigns a single number to quantify the amount of moment magnitude scale#Radiated seismic energy released by an earthquake.... measures earthquake
Earthquake
An earthquake is the result of a sudden release of energy in the Earth's crust that creates seismic waves. Earthquakes are recorded with a seismometer, also known as a seismograph.... intensity on a base-10 logarithmic scale.
Optics is the study of the behavior and properties of light including its optical phenomena with matter and its imaging by optical instruments.... , the absorbance unit used to measure optical density
Optical density
In optics, density is a unitless measure of the transmittance of an optical element for a given length at a given wavelength ?:|||= the per-unit opacity ... is equivalent to -1 B.
Astronomy is the science of Astronomical object and Phenomenon that originate outside the Earth's atmosphere . It is concerned with the evolution, physics, chemistry, meteorology, and motion of celestial objects, as well as the physical cosmology.... , the apparent magnitude
Apparent magnitude
The apparent magnitude of a celestial body is a measurement of its brightness as seen by an observer on Earth, normalized to the value it would have in the absence of the Earth's atmosphere.... measures the brightness of star
Star
A star is a massive, luminous ball of Plasma that is held together by its own gravity. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth.... s logarithmically, since the eye also responds logarithmically to brightness.
Psychophysics is a subdiscipline of psychology dealing with the relationship between physical stimulus and their subjectivity correlates, or percepts.... , the Weber–Fechner law
Weber–Fechner law
The Weber?Fechner law attempts to describe the relationship between the physical magnitudes of Stimulus and the perceived intensity of the stimuli.... proposes a logarithmic relationship between stimulus and sensation.
Computer science is the study of the theoretical foundations of information and computation, and of practical techniques for their implementation and application in computer systems.... , logarithms often appear in bounds for computational complexity
Computational Complexity
Computational Complexity may refer to:*Computational complexity theory*Computational Complexity ... . For example, to sort
Comparison sort
A comparison sort is a type of sorting algorithm that only reads the list elements through a single abstract comparison operation that determines which of two elements should occur first in the final sorted list.... N items using comparison can require time proportional to the product N × log N. Similarly, base-2 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
Natural number
In mathematics, a natural number can mean either an element of the Set = *n = = ? = ? ... N can be represented in no more than (log2N) + 1 bits.
Information theory is a branch of applied mathematics and electrical engineering involving the quantification of information. Historically, information theory was developed by Claude E.... 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 log2Nbit
Bit
A bit is a binary numeral system numerical digit, taking a value of either 0 or 1. Binary digits are a basic unit of information Computer data storage and transmission in digital computing and digital information theory.... s.
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers.... the logarithm is used to form the metric
Metric space
In mathematics, a metric space is a Set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space.... for the half-plane model
Hyperbolic motion
In geometry, a hyperbolic motion is a mapping of a model of hyperbolic geometry that preserves the distance measure in the model. Such a mapping is analogous to congruences of Euclidean geometry which are compositions of rotations and translations.... of hyperbolic geometry
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced. The parallel postulate in Euclidean geometry is equivalent to the statement that, in two dimensional space, for any given line l and point P not on l, there is exactly one line through P th... .
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.
In inferential statistics, the logarithm of the data in a dataset can be used for parametric statistical testing if the original data do not meet the assumption of normality
Normality
Normality may refer to:* The property of conforming to a norm ; see normal , assimilation ;* In chemistry, Concentration#Normality: it is equal to the number of gram equivalents of a solute per liter of solution.... .
In music theory, the term interval describes the relationship between the pitch of two notes.Intervals may be described as:*vertical if the two notes sound simultaneously... are measured logarithmically as semitone
Semitone
A semitone, also called a half step or a half tone,Aaron Copland, Leonard Bernstein, and others use "half tone".One source says that step is "chiefly US", and that half-tone is "chiefly N.... s. The interval between two notes in semitones is the base-21/12 logarithm of the frequency ratio (or equivalently, 12 times the base-2 logarithm). Fractional semitones are used for non-equal temperaments. Especially to measure deviations from the equal tempered scale, intervals are also expressed in cents
Cent (music)
The cent is a logarithmic scale unit of measure used for musical interval . Typically cents are used to measure extremely small intervals, or to compare the sizes of comparable intervals in different tuning systems, and in fact the interval of one cent is much too small to be heard between successive notes.... (hundredths of an equally-tempered semitone
Semitone
A semitone, also called a half step or a half tone,Aaron Copland, Leonard Bernstein, and others use "half tone".One source says that step is "chiefly US", and that half-tone is "chiefly N.... ). The interval between two notes in cents is the base-21/1200 logarithm of the frequency ratio (or 1200 times the base-2 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 ).
The exponential function is a function in mathematics. The application of this function to a value x is written as exp. Equivalently, this can be written in the form ex, where e is the mathematical constant that is the base of the natural logarithm and that is also known as Euler's number.... 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 computer
Computer
A computer is a machine that manipulates Data according to a list of Code .The first devices that resemble modern computers date to the mid-20th century , although the computer concept and various machines similar to computers existed earlier.... s and calculator
Calculator
A calculator is a device for performing mathematical calculations, distinguished from a computer by having a limited problem solving ability and an interface optimized for interactive calculation rather than programming.... 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 interpolating
Interpolation
In the mathematics subfield of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points.... 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 chronometer
Marine chronometer
A marine chronometer is a timekeeper precise enough to be used as a portable time standard; it can therefore be used to determine longitude by means of celestial navigation.... made possible the accurate measurement of longitude
Longitude
Longitude , symbolized by the Greek character lambda , is the geographic coordinate most commonly used in cartography and global navigation for east-west measurement.... 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 rule
Slide rule
The slide rule, also known colloquially as a slipstick, is a mechanical analog computer. The slide rule is used primarily for multiplication and division , and also for "scientific" functions such as Nth roots, logarithms and trigonometry, but does not generally perform addition or subtraction.... , 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 reciprocal of the number: cologb(x) = logb(1/x) = −logb(x). This terminology is found primarily in older books.
In mathematics, if ƒ is a function from A to B then an inverse function for ƒ is a function in the opposite direction, from B to A, with the property that a round trip from A to B to A returns each element of the initial set to itself.... of the logarithm function logb(x); it can be written in closed form as by. The antilog notation was common before the advent of modern calculators and computers: tables of antilogarithms to the base 10 were useful in carrying out computations by hand. The notation still appears in some modern books, and is still used in some situations. For example, certain electronic circuit components are known as antilog amplifiers.
Calculus
The natural logarithm of a positive number x can be defined as
This function is also commonly denoted by log.
This definition satisfies the usual properties of a logarithm. For example, it can be shown as follows that ln(xr) = r ln(x). To see this, consider the definition and the change of variable u := t1/r. Then, by the integration by substitution
Integration by substitution
In calculus, integration by substitution is a tool for finding antiderivatives and integrals. Using the fundamental theorem of calculus often requires finding an antiderivative.... theorem:
Likewise, it can be shown that this function verifies the property ln(xy) = ln(x) + ln(y) using
Using the change of variable u := t/x in the last integral yields
as desired.
Using the last two properties, the rule ln(x / y) = ln(x) − ln(y) can be proved:
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point.... of the natural logarithm function is
By applying the change-of-base rule, the derivative for other bases is
In calculus, an antiderivative, primitive or indefinite integralof a function f is a function F whose derivative is equal to f, i.e., F ′ = f.... of the natural logarithm ln(x) is
In calculus, an antiderivative, primitive or indefinite integralof a function f is a function F whose derivative is equal to f, i.e., F ′ = f.... of the logarithm for other bases is
The following is a list of integrals of logarithmic functions. For a complete list of integral functions, see list of integrals.Note:x>0 is assumed throughout this article, and the constant of integration is omitted for simplicity.... .
Series for calculating the natural logarithm
Basic series
There are several series for calculating natural logarithms. The simplest, though inefficient, is:
To derive this series, start with (|x| < 1)
Integrate both sides to obtain
Letting z = 1 − x and thus x = 1 − z, we get
More efficient series
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
The series converges most quickly if z is close to 1. For high-precision calculations, we can first obtain a low-accuracy approximation y ˜ ln(z), then let A = z/exp(y), where exp(y) can be calculated using the exponential series, which converges quickly provided y is not too large. Then ln(z) = y + ln(A), where A is close to 1 as desired. Larger z can be handled by writing z = a × 10b, whence ln(z) = ln(a) + b × ln(10) (using 10 as an example base). High precision calculations can be first obtained by low accuracy as mentioned above, this helps in the mathematical process.
Example
For example, applying this series to
we get
and thus
where we factored 2/10 out of the sum in the first line.
For any other base b, we use
About convergence
The above series for converges for all complex number , . In fact, as seen by the ratio test
Ratio test
In mathematics, the ratio test is a convergence tests for the convergent series of a series whose terms are real or complex numbers. The test was first published by Jean le Rond d'Alembert and is sometimes known as d'Alembert's ratio test.... , it has radius of convergence
Radius of convergence
In mathematics, the radius of convergence of a power series is a non-negative quantity, either a real number or that represents a domain in which the power series will Convergence.... equal to 1, therefore converges absolutely on every disk
Disk (mathematics)
In geometry, a disk is the region in a plane bounded by a circle.A disk is said to be closed or open according to whether or not it contains the circle that constitutes its boundary.... with radius r<1. Moreover, it converges uniformly on every nibbled disk , with . This follows at once from the algebraic identity:
,
just observing that the right-hand side is uniformly convergent on the whole closed unit disk.
Computers
Many 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
Floating point
In computing, floating point describes a system for numerical representation in which a String of digits represents a rational number.The term floating point refers to the fact that the radix point can "float": that is, it can be placed anywhere relative to the Significant figures of the number.... (or double precision
Double precision
In computing, double precision is a computer numbering format that occupies two adjacent storage locations in computer memory. A double precision number, sometimes simply called a double, may be defined to be an integer, fixed point, or floating point.... ) data type.
In computing, floating point describes a system for numerical representation in which a String of digits represents a rational number.The term floating point refers to the fact that the radix point can "float": that is, it can be placed anywhere relative to the Significant figures of the number.... , it can be useful to consider the following:
A floating point value x is represented by a mantissa
Significand
The significand is the part of a floating point that contains its significant digits. Depending on the interpretation of the exponent, the significand may be considered to be an integer or a fraction .... 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(unsigned 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
Complex number
In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:... arguments, though it is a multivalued function
Multivalued function
In mathematics, a multivalued function is a total relation; i.e. every input is associated with one or more outputs. Strictly speaking, a "well-defined" function associates one, and only one, output to any particular input.... that needs a branch cut terminating at the branch point
Branch point
In the mathematics field of complex analysis, a branch point of a multivalued function is a point such that the function is discontinuous when going around an arbitrarily small circuit around this point .... at 0 to make an ordinary function or principal branch
Principal branch
In mathematics, a principal branch is a function which selects one branch, or "slice", of a multi-valued function. Most often, this applies to functions defined on the complex plane: see branch cut.... . The logarithm (to base e) of a complex number z is the complex number ln(|z|) + i arg(z), where |z| is the modulus
Absolute value
In mathematics, the absolute value of a real number is its numerical value without regard to its Negative and non-negative numbers. So, for example, 3 is the absolute value of both 3 and -3.... of z, arg(z) is the argument
Complex number
In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:... , and i is the imaginary unit
Imaginary unit
In mathematics, physics, and engineering, the imaginary unit is denoted by or the Latin or the Greek iota . It allows the real number system, to be extended to the complex number system, Its precise definition is dependent upon the particular method of extension.... ; see complex logarithm
Complex logarithm
In complex analysis, a complex logarithm function is an "inverse function" of the complex exponential function, just as the natural logarithm ln x is the inverse of the exponential function ex.... for details.
In mathematics, specifically in abstract algebra and its applications, discrete logarithms are group analogues of ordinary logarithms. In particular, an ordinary logarithm loga is a solution of the equation ax = b over the real or complex numbers.... is a related notion in the theory of finite group
Finite group
In mathematics, a finite group is a group that has finite setly many elements. During the twentieth century, mathematicians investigated some aspects of the theory of finite groups in great depth: in particular, the local analysis of finite groups, and the theory of solvable groups and nilpotent groups.... 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 matrix function which generalizes the scalar logarithm to matrix . It is in some sense an inverse function of the matrix exponential.... is the inverse of the matrix exponential
Matrix exponential
In mathematics, the matrix exponential is a matrix function on square matrix analogous to the ordinary exponential function. Abstractly, the matrix exponential gives the connection between a matrix Lie algebra and the corresponding Lie group.... .
It is possible to take the logarithm of a quaternion
Quaternion
Quaternions, in mathematics, are a non-commutative number system that extends the complex numbers. The quaternions were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space.... s and octonion
Octonion
In mathematics, the octonions are a associative extension of the quaternions. Their 8-dimensional normed division algebra over the real numbers is the widest possible that can be obtained from the Cayley-Dickson construction.... s.
A double exponential function is a constant raised to the power of an exponential function. The general formula is , which grows even faster than an exponential function.... . A super-logarithm
Super-logarithm
In mathematics, the super-logarithm is one of the two inverse functions of tetration. Just as exponentiation has two inverse functions: Nth root and logarithms, likewise tetration has two inverse functions: super-roots and super-logarithms.... or hyper
Hyper operator
The hyper operators forming the hypern family are related to Knuth's up-arrow notation and Conway chained arrow notation as follows:... -4-logarithm is the inverse function of tetration
Tetration
In mathematics, tetration is an iterated function exponential function, the first hyper operator after exponentiation. The portmanteau tetration was coined by English mathematician Reuben Louis Goodstein from tetra- and iteration.... . The super-logarithm
Super-logarithm
In mathematics, the super-logarithm is one of the two inverse functions of tetration. Just as exponentiation has two inverse functions: Nth root and logarithms, likewise tetration has two inverse functions: super-roots and super-logarithms.... 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
Isomorphism
In abstract algebra, an isomorphism is a bijection map f such that both f and its inverse function f −1 are homomorphisms, i.e., structure-preserving mappings.... from the group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element.... 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
Haar measure
In mathematical analysis, the Haar measure is a way to assign an "invariant volume" to subsets of locally compact topological groups and subsequently define an integral for functions on those groups.... in the topological group
Topological group
In mathematics, a topological group is a group G together with a topological space on G such that the group's binary operation and the group's inverse function are continuous function .... 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
John Napier
John Napier of Merchistoun - also signed as Neper, Nepair - named Marvellous Merchiston, was a Scotland mathematics, physicist, astronomer/astrologer and 8th Laird of Merchistoun, son of Sir Archibald Napier of Merchiston.... , Baron of Merchiston, in Scotland
Scotland
conventional_long_name = ScotlandAlba|common_name= Scotland|image_flag = Flag of Scotland.svg|flag_width = 130px... . (Joost Bürgi
Joost Bürgi
Joost B?rgi, or Jobst B?rgi ,active primarily at the courts in Kassel, Hesse-Kassel ) and Praha was a Swiss clockmaker, a maker of astronomical instruments and a mathematics.... 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
Johannes Kepler
Johannes Kepler was a Germans mathematician, astronomer and astrologer, and key figure in the 17th century Scientific revolution. He is best known for his eponymous Kepler's laws of planetary motion, codified by later astronomers based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican Astrononomy.... 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
Prosthaphaeresis
Prosthaphaeresis was an algorithm used in the late 16th century and early 17th century for approximate multiplication and Division using formulas from trigonometry.... , 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
Numerical integration
In numerical analysis, numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical ordinary differential equations.... of a hyperbolic sector
Hyperbolic sector
A hyperbolic sector is a region of the Cartesian plane bounded by rays from the origin to two points and and by the hyperbola xy = 1.... at the hand of Gregoire de Saint-Vincent
Grégoire de Saint-Vincent
Gr?goire de Saint-Vincent , a Jesuit, was a mathematician who discovered that the area under a rectangular hyperbola is the same over [a,b] as over [c,d] when a/b = c/d.... 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
Logos
is an important term in philosophy, analytical psychology, rhetoric and religion.Heraclitus established the term in Western philosophy as meaning both the source and fundamental order of the cosmos.... ) 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
Geometric series
In mathematics, a geometric series is a series with a constant ratio between successive term . For example, the seriesis geometric, because each term is equal to half of the previous term.... 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 that manipulates Data according to a list of Code .The first devices that resemble modern computers date to the mid-20th century , although the computer concept and various machines similar to computers existed earlier.... s and calculator
Calculator
A calculator is a device for performing mathematical calculations, distinguished from a computer by having a limited problem solving ability and an interface optimized for interactive calculation rather than programming.... 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 logarithm
Common logarithm
The common logarithm is the logarithm with base 10. It is also known as the decadic logarithm, named after its base. It is indicated by log10, or sometimes Log with a capital L .... for details, including the use of characteristics and mantissa
Mantissa
Mantissa may refer to:* Mantissa * Significand, part of a floating-point number* Part of a common logarithm* A novel by John Fowles... s of common (i.e., base-10) logarithms.
Henry Briggs was an England mathematics notable for changing Napier's logarithms into Common logarithm/Briggesian logarithms.... published the first installment of his own table of common logarithms, containing the logarithms of all integers below 1000 to eight decimal
Decimal
The decimal numeral system has 10 as its Base . It is the most widely used numeral system.... 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
Adriaan Vlacq
Adriaan Vlacq was a Netherlands book publisher and mathematician. Born in Gouda, Vlacq published a table of Henry Briggs logarithms from 1 to 100,000 to 10 decimal places in 1628 in his Arithmetica logarithmica.... , 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 Leipzig
Leipzig
Leipzig is, with a population of over 511,252, the largest city in the States of Germany of Saxony, Germany.... in 1794 under the title Thesaurus Logarithmorum Completus by Jurij Vega
Jurij Vega
Baron Jurij Bartolomej Vega was a Slovenes mathematician, physicist and artillery Commissioned officer.... .
Paris is the Capital of France and the country's largest city. It is situated on the river Seine, in northern France, at the heart of the ?le-de-France Regions of France .... , 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
Interpolation
In the mathematics subfield of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points.... , 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 function
Trigonometric function
In mathematics, the trigonometric functions are function s of an angle. They are important in the trigonometry of Triangle and modeling Periodic function, among many other applications.... s.
Besides the tables mentioned above, a great collection, called Tables du Cadastre, was constructed under the direction of Gaspard de Prony
Gaspard de Prony
Gaspard Clair Fran?ois Marie Riche de Prony was a France mathematician and engineer, who worked on hydraulics. He was born at Chamelet, Beaujolais, France.... , by an original computation, under the auspices of the French
France
France , officially the French Republic , is a country whose Metropolitan France is located in Western Europe and that also comprises various Overseas departments and territories of France.... 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." Cubic
Cubic function
In mathematics, a cubic function is a function of the formwhere a is nonzero; or in other words, a polynomial of Degree of a polynomial three.... interpolation
Interpolation
In the mathematics subfield of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points.... could be used to find the logarithm of any number to a similar accuracy.
MathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by Wolfram Research Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at Urbana-Champaign....
