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Logarithm functions, graphed for various bases: red is to base e, green is to base 10, and purple is to base 1.7. Each tick on the axes is one unit. Logarithms of all bases pass through the point (1, 0), because any number raised to the power 0 is 1, and through the points (b, 1) for base b, because a number raised to the power 1 is itself. The curves approach the y-axis but do not reach it because of the singularity at x = 0.
Logarithm functions, graphed for various bases: red is to base e, green is to base 10, and purple is to base 1. 7. Each tick on the axes is one unit. Logarithms of all bases pass through the point (1, 0), because any number raised to the power 0 is 1, and through the points (b, 1) for base b, because a number raised to the power 1 is itself. The curves approach the y-axis but do not reach it because of the singularity at x = 0. In Mathematics, a singularity is in general a point at which a given mathematical object is not defined or a point of an exceptional set where it fails to be
The 1797 Britannica explains logarithms as "a series of numbers in arithmetical progression, corresponding to others in geometrical progression; by means of which, arithmetical calculations can be made with much more ease and expedition than otherwise."
The 1797 Britannica explains logarithms as "a series of numbers in arithmetical progression, corresponding to others in geometrical progression; by means of which, arithmetical calculations can be made with much more ease and expedition than otherwise. "

In mathematics, the logarithm of a given number to a given base is the power or exponent to which the base must be raised in order to produce the given number. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and radix|basis (topologyIn Arithmetic, the base refers to the number b in an expression of the form b n.

For example, the logarithm of 1000 to the common base 10 is 3, because 10 raised to the power of 3 is 1000; the base 2 logarithm of 32 is 5 because 2 to the power 5 is 32.

The logarithm of x to the base b is written logb(x) or, if the base is implicit, as log(x). So, for a number x, a base b and an exponent y,

\mbox{if}~~ b^y = x, ~~\mbox{then}~~ \log_b (x) = y \,.

An important feature of logarithms is that they reduce multiplication to addition, by the formula:

\log (x \times y) = \log x + \log y \,.

That is, the logarithm of the product of two numbers is the sum of the logarithms of those numbers. The use of logarithms to facilitate complex calculations was a significant motivation in their original development.

Contents

Properties of the logarithm

Main article: List of logarithmic identities

When x and b are restricted to positive real numbers, logb(x) is a unique real number. In Mathematics, there are several logarithmic identities. Algebraic identities Using simpler operations Logarithms can be used In Mathematics, the real numbers may be described informally in several different ways The magnitude of the base b must be neither 0 nor 1; the base used is typically 10, e, or 2. In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign. The Mathematical constant e is the unique Real number such that the function e x has the same value as the slope of the tangent line Logarithms are defined for real numbers and for complex numbers. [1][2]

The major property of logarithms is that they map multiplication to addition. This ability stems from the following identity:

b^x \times b^y = b^{x+y} \ ,

which by taking logarithms becomes

\log_b \left(b^x \times b^y \right) = \log_b \left( b^{x+y} \right) \ = x + y = \log_b \left(b^x \right) + \log_b \left(b^y \right). \

A related property is reduction of exponentiation to multiplication. Using the identity:

c = b^{\log_b (c )} \ ,

it follows that c to the power p (exponentiation) is:

c^p = \left(b^{\log_b (c )}\right)^p = b^{p \log_b (c )} \ ,

or, taking logarithms:

\log_b \left(c^p \right) = p \log_b (c ) \ .

In words, to raise a number to a power p, find the logarithm of the number and multiply it by p. The exponentiated value is then the inverse logarithm of this product; that is, number to power = bproduct.

Besides reducing multiplication operations to addition, and exponentiation to multiplication, logarithms reduce division to subtraction, and roots to division. Logarithms make lengthy numerical operations easier to perform. 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 whole process is made easy by using tables of logarithms, or a slide rule, antiquated now that calculators are available. 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 slide rule, also known as a slipstick, is a mechanical Analog computer. Although the above practical advantages are not important for numerical work today, they are used in graphical analysis (see Bode plot). A Bode plot, named after Hendrik Wade Bode, is usually a combination of a Bode magnitude plot and Bode phase plot A Bode magnitude plot is a graph of log

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. This article is about both real and complex analytic 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 b is fixed and so the only argument is x. radix|basis (topologyIn Arithmetic, the base refers to the number b in an expression of the form b n. Thus there is one logarithm function for each value of the base b (which must be positive and must differ from 1). Viewed in this way, the base-b logarithm function is the inverse function of the exponential function bx. In Mathematics, if &fnof is a function from A to B then an inverse function for &fnof is a function in the opposite direction from B The word "logarithm" is often used to refer to a logarithm function itself as well as to particular values of this function.

The base can also be a complex number; the evaluation of the log is just slightly more complicated in this case. See imaginary base. In Complex analysis, the complex logarithm is the extension of the Natural logarithm function ln( x) &ndash originally defined for Real numbers

Logarithm of a complex number

When the base b is real and z is a complex number, say z = x + i y, the logarithm of z is found easily by putting z in polar form that is, z = (x2 + y2)1/2 exp (i tan−1 (y / x) ). Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted If the base of the logarithm is chosen as e [3], that is, using loge (denoted by ln and called the natural logarithm), the logarithm becomes:

\ln(z) = \ln \left[ \left( x^2 + y^2 \right) ^{1/2} e^{i \tan^{-1}y/x}\right]
= \ln \left[ \left( x^2 + y^2 \right) ^{1/2}\right] + \ln \left[ e^{i \tan^{-1}y/x}\right]
=\frac {1}{2} \ln \left( x^2 + y^2 \right) + i \tan^{-1}\left( \frac {y} {x} \right) \ .

This evaluation uses the properties of all logarithms (see above), regardless of choice of base: logb (c d ) = logb (c ) + logb (d ) and its generalization to arbitrary products logb bz = z. 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 Because the inverse tangent is a multiple valued function of its argument, the logarithm of a complex number is not unique either. See article on complex logarithm. In Complex analysis, the complex logarithm is the extension of the Natural logarithm function ln( x) &ndash originally defined for Real numbers

Group theory

From the pure mathematical perspective, the identity

\log(cd) = \log(c) + \log(d) \,

is fundamental in two senses. First, the remaining three arithmetic properties can be derived from it. Furthermore, it expresses an isomorphism between the multiplicative group of the positive real numbers and the additive group of all the reals. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Mathematics and Group theory the term multiplicative group refers to one of the following concepts depending on the context any group \scriptstyle\mathfrak In Mathematics, an additive group may be an Abelian group, when it is written using the symbol + for its Binary operation

Logarithmic functions are the only continuous isomorphisms from the multiplicative group of positive real numbers to the additive group of real numbers.

Bases

The most widely used bases for logarithms are 10, the mathematical constant e ≈ 2. The Mathematical constant e is the unique Real number such that the function e x has the same value as the slope of the tangent line 71828. . . and 2. When "log" is written without a base (b missing from logb), the intent can usually be determined from context:

To avoid confusion, it is best to specify the base if there is any chance of misinterpretation.

Other notations

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

This chaos, historically, originates from the fact that the natural logarithm has nice mathematical properties (such as its derivative being 1/x, and having a simple definition), while the base 10 logarithms, or decimal logarithms, were more convenient for speeding calculations (back when they were used for that purpose). Thus natural logarithms were only extensively used in fields like calculus while decimal logarithms were widely used elsewhere.

As recently as 1984, Paul Halmos in his "automathography" I Want to Be a Mathematician heaped contempt on what he considered the childish "ln" notation, which he said no mathematician had ever used. Paul Richard Halmos ( March 3 1916 &mdash October 2 2006) was a Hungarian -born Jewish American Mathematician The notation was in fact invented in 1893 by Irving Stringham, professor of mathematics at Berkeley. The University of California Berkeley (also referred to as Cal, Berkeley and UC Berkeley) is a major research university located in Berkeley As of 2005, many mathematicians have adopted the "ln" notation, but most use "log". Year 2005 ( MMV) was a Common year starting on Saturday (link displays full calendar of the Gregorian calendar.

In computer science, the base 2 logarithm is sometimes written as lg(x), as suggested by Edward Reingold and popularized by Donald Knuth. Edward M Reingold is a Computer scientist active in the fields of Algorithms Data structures and Calendrical calculations He has co-authored the Donald Ervin Knuth (kəˈnuːθ (born 10 January 1938) is a renowned computer scientist and Professor Emeritus of the Art of Computer However, lg(x) is also sometimes used for the common log, and lb(x) for the binary log. [5] In Russian literature, the notation lg(x) is also generally used for the base 10 logarithm. [6] In German, lg(x) also denotes the base 10 logarithm, while sometimes ld(x) or lb(x) is used for the base 2 logarithm.

The clear advice of the United States Department of Commerce National Institute of Standards and Technology is to follow the ISO standard Mathematical signs and symbols for use in physical sciences and technology, ISO 31-11:1992, which suggests these notations:[7]

As the difference between logarithms to different bases is one of scale, it is possible to consider all logarithm functions to be the same, merely giving the answer in different units, such as dB, neper, bits, decades, etc. ; see the section Science and engineering below. Logarithms to a base less than 1 have a negative scale, or a flip about the x axis, relative to logarithms of base greater than 1.

Change of base

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

\log_b(x) = \frac{\log_k(x)}{\log_k(b)}.

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

\log_2(16) = \frac{\log(16)}{\log(2)}.

Uses of logarithms

Logarithms are useful in solving equations in which exponents are unknown. They have simple derivatives, so they are often used in the solution of integrals. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space The logarithm is one of three closely related functions. In the equation bn = x, b can be determined with radicals, n with logarithms, and x with exponentials. In Mathematics, an n th root of a Number a is a number b such that bn = a. The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) See logarithmic identities for several rules governing the logarithm functions. In Mathematics, there are several logarithmic identities. Algebraic identities Using simpler operations Logarithms can be used

Science

Various quantities in science are expressed as logarithms of other quantities; see logarithmic scale for an explanation and a more complete list. Definition and base Logarithmic scales are either defined for ratios of the underlying quantity or one has to agree to measure

Exponential functions

One way of defining the exponential function ex, also written as exp(x), is as the inverse of the natural logarithm. The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) It is positive for every real argument x.

The operation of "raising b to a power p" for positive arguments b and all real exponents p is defined by

b^p = \left( e^{\ln b} \right) ^p = e^{p \ln b }.\,

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:

Operation with numbers Operation with exponents Logarithmic identity
\!\, c d \!\, C + D \!\, \log(c d) = \log(c) + \log(d)
\!\, c / d \!\, C - D \!\, \log(c / d) = \log(c) - \log(d)
\!\, c ^ d \!\, Cd \!\, \log(c ^ d) = d \log(c)
\!\, \sqrt[d]{c} \!\, C / d \!\, \log(\sqrt[d]{c}) = \frac{\log(c)}{d}

These arithmetic properties of logarithms make such calculations much faster. In Mathematics, there are several logarithmic identities. Algebraic identities Using simpler operations Logarithms can be used The use of logarithms was an essential skill until electronic computers and calculators became available. A computer is a Machine that manipulates data according to a list of instructions. A calculator is device for performing mathematical calculations distinguished from a Computer by having a limited problem solving ability and an interface optimized for interactive Indeed the discovery of logarithms, just before Newton's era, had an impact in the scientific world that can be compared with that of the advent of computers in the 20th century because it made feasible many calculations that had previously been too laborious.

As an example, to approximate the product of two numbers one can look up their logarithms in a table, add them, and, using the table again, proceed from that sum to its antilogarithm, which is the desired product. The precision of the approximation can be increased by interpolating between table entries. In the mathematical subfield of Numerical analysis, interpolation is a method of constructing new data points within the range of a Discrete set of 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.

The C and D scales on this slide rule are marked off at positions corresponding to the logarithms of the numbers shown. By mechanically adding the logs of 1.3 and 2, the cursor shows the product is 2.6.
The C and D scales on this slide rule are marked off at positions corresponding to the logarithms of the numbers shown. The slide rule, also known as a slipstick, is a mechanical Analog computer. By mechanically adding the logs of 1. 3 and 2, the cursor shows the product is 2. 6.

One key application of these techniques was celestial navigation. Once the invention of the chronometer made possible the accurate measurement of longitude at sea, mariners had everything necessary to reduce their navigational computations to mere additions. A 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 Longitude (ˈlɒndʒɪˌtjuːd or ˈlɒŋgɪˌtjuːd symbolized by the Greek character Lambda (λ is the east-west Geographic coordinate measurement 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, an essential calculating tool for engineers. The slide rule, also known as a slipstick, is a mechanical Analog computer. 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). [8]

Antilogarithms

The antilogarithm is the logarithmic inverse of the logarithm, meaning that the antilogb(logb(x))=x. Thus, setting by=x implies that logb(x)=y. By taking the antilogb of both sides, antilogb(logb(x))=antilogby, thus x=antilogby. Therefore, by=antilogby.

Calculus

The natural logarithm of a positive number x can be defined as

\ln (x) \equiv \int_{1}^{x} \frac{dt}{t}.

The derivative of the natural logarithm function is

\frac{d}{dx} \ln(x) = \frac{1}{x}.

By applying the change-of-base rule, the derivative for other bases is

\frac{d}{dx} \log_b(x) = \frac{d}{dx} \frac {\ln(x)}{\ln(b)} = \frac{1}{x \ln(b)} = \frac{\log_b(e)}{x}.

The antiderivative of the natural logarithm ln(x) is

\int \ln(x) \,dx = x \ln(x) - x + C,

and so the antiderivative of the logarithm for other bases is

\int \log_b(x) \,dx = x \log_b(x) - \frac{x}{\ln(b)} + C = x \log_b \left(\frac{x}{e}\right) + C.

See also: Table of limits, list of integrals of logarithmic functions. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In Calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose Derivative In Calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose Derivative This is a table of limits for common functions Note that a and b are constants with respect to x. The following is a list of Integrals ( Antiderivative functions of Logarithmic functions For a complete list of integral functions see List of integrals

Series for calculating the natural logarithm

There are several series for calculating natural logarithms. [9] The simplest, though inefficient, is:

\ln (z) = \sum_{n=1}^\infty \frac{-{(-1)}^n}{n} (z-1)^n when |z-1|<1 \!.

To derive this series, start with (|x|<1 \!.)

\frac{1}{1-x} = 1 + x + x^2 + x^3 + \cdots.

Integrate both sides to obtain

-\ln(1-x) = x + \frac{x^2}{2} + \frac{x^3}{3} + \cdots
\ln(1-x) = -x - \frac{x^2}{2} - \frac{x^3}{3} - \frac{x^4}{4} - \cdots.

Letting z = 1-x \! and thus x = -(z-1) \!, we get

\ln z = (z-1) - \frac{(z-1)^2}{2} + \frac{(z-1)^3}{3} - \frac{(z-1)^4}{4} + \cdots

A more efficient series is

\ln (z) = 2 \sum_{n=0}^\infty \frac{1}{2n+1} {\left ( \frac{z-1}{z+1} \right ) }^{2n+1}

for z with positive real part.

To derive this series, we begin by substituting −x for x and get

\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots.

Subtracting, we get

\ln \frac{1+x}{1-x} = \ln(1+x) - \ln(1-x) = 2x + 2\frac{x^3}{3} + 2\frac{x^5}{5} + \cdots.

Letting z = \frac{1+x}{1-x} \! and thus x = \frac{z-1}{z+1} \!, we get

\ln z = 2 \left ( \frac{z-1}{z+1} + \frac{1}{3}{\left(\frac{z-1}{z+1}\right)}^3 + \frac{1}{5}{\left(\frac{z-1}{z+1}\right)}^5 + \cdots \right ).

For example, applying this series to

z = \frac{11}{9},

we get

\frac{z-1}{z+1} = \frac{\frac{11}{9} - 1}{\frac{11}{9} + 1} = \frac{1}{10},

and thus

\ln (1.2222222\dots) = \frac{2}{10} \left (1 + \frac{1}{3\cdot 100} + \frac{1}{5 \cdot 10000} + \frac{1}{7 \cdot 1000000} + \cdots \right )
= 0.2 \cdot (1.0000000\dots + 0.0033333\dots + 0.0000200\dots + 0.0000001\dots + \cdots)
= 0.2 \cdot 1.0033535\dots = 0.2006707\dots

where we factored 1/10 out of the sum in the first line.

For any other base b, we use

\log_b (x) = \frac{\ln (x)}{\ln (b)}.

Computers

Most computer languages use log(x) for the natural logarithm, while the common log is typically denoted log10(x). The argument and return values are typically a floating point (or double precision) data type. In Computing, floating point describes a system for numerical representation in which a string of digits (or Bits represents a Real number. In Computing, double precision is a Computer numbering format that occupies two adjacent storage locations in computer memory

As the argument is floating point, it can be useful to consider the following:

A floating point value x is represented by a mantissa m and exponent n to form

x = m2^n.\,

Therefore

\ln(x) = \ln(m) + n\ln(2).\,

Thus, instead of computing ln(x) we compute ln(m) for some m such that 1 ≤ m <  2. In Computing, floating point describes a system for numerical representation in which a string of digits (or Bits represents a Real number. The significand (also Coefficient or Mantissa) is the part of a floating-point number that contains its significant digits Having m in this range means that the value u = \frac{m - 1}{m+1} is always in the range 0 \le u < \frac13. Some machines use the mantissa in the range 0.5 \le m < 1 and in that case the value for u will be in the range -\frac13 < u \le 0 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){
  int r = 0;
  while( (x >> r) != 0){
    r++;
  }
  return r-1; // returns -1 for x==0, floor(log2(x)) otherwise
}

However, it can also be computed in O(log n) steps by trying to shift by powers of 2 and checking that the result stays nonzero: for example, first >>16, then >>8, . . . (Each step reveals one bit of the result)

Generalizations

The ordinary logarithm of positive reals generalizes to negative and complex arguments, though it is a multivalued function that needs a branch cut terminating at the branch point at 0 to make an ordinary function or principal branch. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, a multivalued function (shortly multifunction, other names set-valued function, set-valued map, multi-valued map In the mathematical field of Complex analysis, a branch point may be informally thought of as a point z 0 at which a " multi-valued 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 modulus of z, arg(z) is the argument, and i is the imaginary unit; see complex logarithm for details. In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted Definition By definition the imaginary unit i is one solution (of two of the Quadratic equation In Complex analysis, the complex logarithm is the extension of the Natural logarithm function ln( x) &ndash originally defined for Real numbers

The discrete logarithm is a related notion in the theory of finite groups. In Mathematics, specifically in Abstract algebra and its applications discrete logarithms are group-theoretic analogues of ordinary Logarithms In Mathematics, a finite group is a group which has finitely many elements 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. Public-key cryptography, also known as asymmetric cryptography, is a form of Cryptography in which the key used to encrypt a message differs from the key

The logarithm of a matrix is the inverse of the matrix exponential. In Mathematics, the logarithm of a matrix is a Matrix function which generalizes the scalar Logarithm to matrices. In Mathematics, the matrix exponential is a Matrix function on square matrices analogous to the ordinary Exponential function.

It is possible to take the logarithm of a quaternions and octonions. Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician In Mathematics, the octonions are a nonassociative extension of the Quaternions Their 8-dimensional Normed division algebra over the Real

A double logarithm, ln(ln(x)), is the inverse function of the double exponential function. A double exponential function is a Constant raised to the power of an Exponential function. A super-logarithm or hyper-4-logarithm is the inverse function of tetration. In Mathematics, the super-logarithm is one of the two inverse functions of Tetration. The hyper operators forming the hyper n family are related to Knuth's up-arrow notation and Conway chained arrow notation as follows In Mathematics, tetration (also known as hyper -4 The super-logarithm of x grows even more slowly than the double logarithm for large x. In Mathematics, the super-logarithm is one of the two inverse functions of Tetration.

For each positive b not equal to 1, the function logb  (x) is an isomorphism from the group of positive real numbers under multiplication to the group of (all) real numbers under addition. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element They are the only such isomorphisms that are continuous. The logarithm function can be extended to a Haar measure in the topological group of positive real numbers under multiplication. In Mathematical analysis, the Haar measure is a way to assign an "invariant volume" to subsets of Locally compact topological groups and subsequently define In Mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the

History

A more modern definition and explanation from 1866 A Dictionary of Science, Literature, & Art: Comprising the Definitions and Derivations of the Scientific Terms in General Use, together with the History and Descriptions of the Scientific Principles of Nearly Every Branch of Human Knowledge
A more modern definition and explanation from 1866 A Dictionary of Science, Literature, & Art: Comprising the Definitions and Derivations of the Scientific Terms in General Use, together with the History and Descriptions of the Scientific Principles of Nearly Every Branch of Human Knowledge

The method of logarithms was first publicly propounded in 1614, in a book entitled Mirifici Logarithmorum Canonis Descriptio, by John Napier, Baron of Merchiston, in Scotland. For other people with the same name see John Napier (disambiguation. Scotland ( Gaelic: Alba) is a Country in northwest Europethat occupies the northern third of the island of Great Britain. [10] (Joost Bürgi independently discovered logarithms; however, he did not publish his discovery until four years after Napier. Joost Bürgi, or Jobst Bürgi ( February 28 1552, Lichtensteig, Switzerland - January 31 1632)active primarily ) Early resistance to the use of logarithms was muted by Kepler's enthusiastic support and his publication of a clear and impeccable explanation of how they worked. Johannes Kepler (ˈkɛplɚ ( December 27 1571 &ndash November 15 1630) was a German Mathematician, Astronomer [11]

Their use contributed to the advance of science, and especially of astronomy, by making some difficult calculations possible. Prior to the advent of calculators and computers, they were used constantly in surveying, navigation, and other branches of practical mathematics. It supplanted the more involved method of prosthaphaeresis, which relied on trigonometric identities as a quick method of computing products. Prosthaphaeresis was an Algorithm used in the late 16th century and early 17th century for approximate Multiplication and division using formulas from In Mathematics, trigonometric identities are equalities that involve Trigonometric functions that are true for every single value of the occurring variables Besides the utility of the logarithm concept in computation, the natural logarithm presented a solution to the problem of quadrature of a hyperbolic sector at the hand of Gregoire de Saint-Vincent in 1647. In Numerical analysis, numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite Integral, and by extension A hyperbolic sector is a region of the Cartesian plane {( x, y)} bounded by rays from the origin to two points ( a, 1/ a) and ( Grégoire de Saint-Vincent ( March 22 1584 Bruges – June 5 1667 Ghent) a Jesuit, was a Mathematician

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: λόγος (logos) meaning proportion, and ἀριθμός (arithmos) meaning number. grc-Latn Logos (ˈloʊːgɒs ( Greek, logos) is an important term in Philosophy, Analytical psychology, Rhetoric and Religion Napier chose that because the difference of two logarithms determines the ratio of the numbers they represent, so that an arithmetic series of logarithms corresponds to a geometric series of numbers. In Mathematics, an arithmetic progression or arithmetic sequence is a Sequence of Numbers such that the difference of any two successive members In Mathematics, a geometric series is a series with a constant ratio between successive terms. The term antilogarithm was introduced in the late 17th century and, while never used extensively in mathematics, persisted in collections of tables until they fell into disuse.

Napier did not use a base as we now understand it, but his logarithms were, up to a scaling factor, effectively to base 1/e. For interpolation purposes and ease of calculation, it is useful to make the ratio r in the geometric series close to 1. Napier chose r = 1 - 10−7 = 0. 999999 (Bürgi chose r = 1 + 10−4 = 1. 0001). Napier's original logarithms did not have log 1 = 0 but rather log 107 = 0. Thus if N is a number and L is its logarithm as calculated by Napier, N = 107(1 − 10−7)L. Since (1 − 10−7)107 is approximately 1/e, this makes L/107 approximately equal to log1/e N/107. [5]

Tables of logarithms

Part of a 20th century table of common logarithms in the reference book Abramowitz and Stegun.
Part of a 20th century table of common logarithms in the reference book Abramowitz and Stegun. The common logarithm is the Logarithm with base 10 It is also known as the decadic logarithm, named after its base Abramowitz and Stegun is the informal name of a mathematical reference work edited by Milton Abramowitz and Irene Stegun of the U

Prior to the advent of computers and calculators, using logarithms meant using tables of logarithms, which had to be created manually. A computer is a Machine that manipulates data according to a list of instructions. A calculator is device for performing mathematical calculations distinguished from a Computer by having a limited problem solving ability and an interface optimized for interactive Base-10 logarithms are useful in computations when electronic means are not available. See common logarithm for details, including the use of characteristics and mantissas of common (i. The common logarithm is the Logarithm with base 10 It is also known as the decadic logarithm, named after its base e. , base-10) logarithms.

In 1617, Henry Briggs published the first installment of his own table of common logarithms, containing the logarithms of all integers below 1000 to eight decimal places. Henry Briggs (February 1561&ndash January 26 1630) was an English mathematician notable for changing Napier's logarithms into The decimal ( base ten or occasionally denary) Numeral system has ten as its base. This he followed, in 1624, by his Arithmetica Logarithmica, containing the logarithms of all integers from 1 to 20,000 and from 90,000 to 100,000 to fourteen places of decimals, together with a learned introduction, in which the theory and use of logarithms are fully developed. The interval from 20,000 to 90,000 was filled up by Adriaan Vlacq, a Dutch mathematician; but in his table, which appeared in 1628, the logarithms were given to only ten places of decimals. Adriaan Vlacq ( Gouda, 1600 - The Hague, 1667 was a Dutch book publisher and mathematician The Netherlands ( Dutch:, ˈnedərlɑnt is the European part of the Kingdom of the Netherlands, which consists of the Netherlands the Netherlands

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. "[12] An edition of Vlacq's work, containing many corrections, was issued at Leipzig in 1794 under the title Thesaurus Logarithmorum Completus by Jurij Vega. This sort of fix restores section edit linkpoints to where they belong Year 1794 ( MDCCXCIV) was a Common year starting on Wednesday (link will display the full calendar of the Gregorian calendar (or a Baron Jurij Bartolomej Vega (also correct Veha; official Georgius Bartholomaei Vecha Georg Freiherr von Vega ( March 23, 1754 &ndash

François Callet's seven-place table (Paris, 1795), instead of stopping at 100,000, gave the eight-place logarithms of the numbers between 100,000 and 108,000, in order to diminish the errors of interpolation, which were greatest in the early part of the table; and this addition was generally included in seven-place tables. Paris (ˈpærɨs in English; in French) is the Capital of France and the country's largest city Year 1795 ( MDCCXCV) was a Common year starting on Thursday (link will display the full calendar of the Gregorian calendar (or a In the mathematical subfield of Numerical analysis, interpolation is a method of constructing new data points within the range of a Discrete set of 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. Year 1871 ( MDCCCLXXI) was a Common year starting on Sunday (link will display the full calendar of the Gregorian calendar (or a Common

Briggs and Vlacq also published original tables of the logarithms of the trigonometric functions.

Besides the tables mentioned above, a great collection, called Tables du Cadastre, was constructed under the direction of Gaspard de Prony, by an original computation, under the auspices of the French republican government of the 1700s. Gaspard Clair François Marie Riche de Prony ( July 22, 1755 - July 29, 1839) was a French Mathematician and Engineer This article is about the country For a topic outline on this subject see List of basic France topics. 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. Year 1792 ( MDCCXCII) was a Leap year starting on Sunday (link will display the full calendar of the Gregorian calendar (or a Leap year " [13] Cubic interpolation could be used to find the logarithm of any number to a similar accuracy. This article discusses cubic equations in one variable For a discussion of cubic equations in two variables see Elliptic curve. In the mathematical subfield of Numerical analysis, interpolation is a method of constructing new data points within the range of a Discrete set of

See also

References

  1. ^ In general, x and b both can be complex numbers; see Kwok below, and imaginary-base logarithms. This is a list of Logarithm topics, by Wikipedia page See also the List of exponential topics. In Mathematics, there are several logarithmic identities. Algebraic identities Using simpler operations Logarithms can be used Definition and base Logarithmic scales are either defined for ratios of the underlying quantity or one has to agree to measure The natural logarithm, formerly known as the Hyperbolic logarithm is the Logarithm to the base e, where e is an irrational The common logarithm is the Logarithm with base 10 It is also known as the decadic logarithm, named after its base In Complex analysis, the complex logarithm is the extension of the Natural logarithm function ln( x) &ndash originally defined for Real numbers In Complex analysis, the complex logarithm is the extension of the Natural logarithm function ln( x) &ndash originally defined for Real numbers The indefinite logarithm of a positive number n (variously denoted n \mathrm{Log}(n or even sometimes just \log n is the logarithm In Computer science, the iterated logarithm of n, written log* n (usually read " log star " is the number of times the In Mathematics, specifically in Abstract algebra and its applications discrete logarithms are group-theoretic analogues of ordinary Logarithms Zech's logarithms are used with Finite fields to reduce a high- degree polynomial that is not in the field to an element in the field (thus having a lower degree In Mathematics, the logarithm of a matrix is a Matrix function which generalizes the scalar Logarithm to matrices. In Probability and Statistics, the log-normal distribution is the single-tailed Probability distribution of any Random variable whose The decibel ( dB) is a logarithmic unit of measurement that expresses the magnitude of a physical quantity (usually power or intensity relative to Equal temperament is a Musical temperament, or a system of tuning in which every pair of adjacent notes has an identical Frequency ratio. The Richter magnitude scale, or more correctly local magnitude M L scale assigns a single number to quantify the amount of seismic energy released pH is the measure of the acidity or alkalinity of a Solution. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Complex analysis, the complex logarithm is the extension of the Natural logarithm function ln( x) &ndash originally defined for Real numbers
  2. ^ Yue Kuen Kwok (2002). Applied complex variables for scientists and engineers. Cambridge MA: Cambridge University Press, p. 102. ISBN 0521004624.  
  3. ^ See e (mathematical constant)
  4. ^ "Mathematisches Lexikon" at Mateh_online.at. The Mathematical constant e is the unique Real number such that the function e x has the same value as the slope of the tangent line
  5. ^ a b Gullberg, Jan (1997). Mathematics: from the birth of numbers. . W. W. Norton & Co. ISBN 039304002X.  
  6. ^ "Common Logarithm" at MathWorld.
  7. ^ B. N. Taylor (1995). Guide for the Use of the International System of Units (SI). NIST Special Publication 811, 1995 Edition. US Department of Commerce.
  8. ^ | Wooster Woodruff B, Smith David E: "Academic Algebra", page 360. Ginn & Company, 1902
  9. ^ Handbook of Mathematical Functions, National Bureau of Standards (Applied Mathematics Series no. 55), June 1964, page 68.
  10. ^ Much of the history of logarithms is derived from The Elements of Logarithms with an Explanation of the Three and Four Place Tables of Logarithmic and Trigonometric Functions, by James Mills Peirce, University Professor of Mathematics in Harvard University, 1873. Year 1873 ( MDCCCLXXIII) was a Common year starting on Wednesday (link will display the full calendar of the Gregorian calendar (or a Common
  11. ^ http://turnbull.dcs.st-and.ac.uk/~history/Biographies/Kepler.html (section "Astronomical Tables")
  12. ^ Athenaeum, 15 June 1872. Events 763 BC - Assyrians record a Solar eclipse that will be used to fix the Chronology of Mesopotamian history Year 1872 ( MDCCCLXXII) was a Leap year starting on Monday (link will display the full calendar of the Gregorian Calendar (or a Leap year See also the Monthly Notices of the Royal Astronomical Society for May 1872. Year 1872 ( MDCCCLXXII) was a Leap year starting on Monday (link will display the full calendar of the Gregorian Calendar (or a Leap year
  13. ^ English Cyclopaedia, Biography, Vol. IV. , article "Prony. "

External links

MathWorld is an online Mathematics reference work created and largely written by Eric W

Dictionary

logarithm

-noun

  1. (mathematics) For a number <math>x</math>, the power to which a given base number must be raised in order to obtain <math>x</math>. Written <math>\log_b x</math>. For example, <math>\log_{10} 1000 = 3</math> because <math>10^3 = 1000</math>.
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