Logarithm

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Logarithm - Wikipedia, the free encyclopedia
Logarithm functions, graphed for various bases: red is to base e, green is to ... In mathematics, the logarithm of a number to a given base is the power or ...
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logarithm: Definition from Answers.com
logarithm n. Mathematics. The power to which a base, such as 10, must be raised to produce a given number ... Britannica Concise Encyclopedia: logarithm ...
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Logarithm -- from Wolfram MathWorld
For any base, the logarithm function has a singularity at. ... Note that while logarithm base 10 is denoted in this work, on calculators, and ...
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Logarithm
history of computing ... However, there is one man who is responsible for the invention of the logarithm. ... The logarithm, therefore, of any sine is a number ...
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natural logarithm: Definition from Answers.com
natural logarithm n. ( Symbol ln ) A logarithm in which the base is the irrational number e (= 2.71828 ... simple terms, the natural logarithm of a number x is ...
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Logarithm - New World Encyclopedia
In mathematics, the logarithm (or log) of a number x in base b is the power (n) ... Or, the logarithm of 81 to the base 3 is 4, because 3 raised to the power ...
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Natural logarithm - Wikipedia, the free encyclopedia
In simple terms, the natural logarithm of a number x is the power to which e ... 6 The natural logarithm in integration. 7 Numerical value. 7.1 High precision ...
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, 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 Mathematical singularity at x = 0.

In mathematics, a logarithm (to base (mathematics) b) of a number x is the exponent y that satisfies x = by. It is written logb(x) or, if the base is implicit, as log(x).

In other words

y=\log_b(x)\,\!

is equivalent to

x= b^y\,\!

The base b must be neither 0 nor 1 (nor a Nth root of 1 in the case of the extension to Complex_number, the complex logarithm), and is typically 10, e (mathematical constant), or 2. When x and b are further restricted to positive real numbers, logb(x) is a unique real number.

For example, since

3^4 = 3 \times 3 \times 3 \times 3 = 81, \,

\log_3(81) = 4, \,

or, in words, the base-3 logarithm of 81 is 4, or the log base-3 of 81 is 4.

The logarithm as a function 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 (mathematics) b is fixed and so the only argument is x. Thus there is one logarithm function for each value of the base b (which must be positive and must differ from 1).Viewed in this way, the base-b logarithm function is the inverse function of the exponentiation bx. The word "logarithm" is often used to refer to a logarithm function itself as well as to particular values of this function.

Integer and non-integer exponents If n is a negative and non-negative numbers integer, bn signifies the multiplication of n factors equal to b:

\underbrace{b \times b \times \cdots \times b}_n.

However, if b is a positive real number not equal to 1, this definition can be extended to any real number n in a field (mathematics) (see exponentiation). Similarly, the logarithm function can be defined for any positive real number. For each positive base b not equal to 1, there is one logarithm function (mathematics) and one exponential function, which are inverses of each other.

Logarithms can reduce multiplication operations to addition, division to subtraction, exponentiation to multiplication, and roots to division. Therefore, logarithms are useful for making lengthy numerical operations easier to perform and, before the advent of electronic computers, they were widely used for this purpose in fields such as astronomy, engineering, navigation, and cartography. They have important mathematical properties and are still widely used today.

Bases The most widely used bases for logarithms are 10, the mathematical constant e (mathematical constant) ≈ 2.71828... 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:













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

As recently as 1984, Paul Halmos in his "automathography" I Want to Be a Mathematician heaped contempt on what he considered the childish "ln" notation, which he said no mathematician had ever used. The notation was in fact invented in 1893 by Irving Stringham, professor of mathematics at University of California, Berkeley. As of 2005, many mathematicians have adopted the "ln" notation, but most use "log".

In computer science, the base 2 logarithm is sometimes written as lg(x), as suggested by Edward Reingold and popularized by Donald Knuth. However, lg(x) is also sometimes used for the common log, and lb(x) for the binary log. In Russian literature, the notation lg(x) is also generally used for the base 10 logarithm.{{cite web] National Institute of Standards and Technology is to follow the International Organization for Standardization standard Mathematical signs and symbols for use in physical sciences and technology, ISO 31-11:1992, which suggests these notations:

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 or bytes; see the section #Science and engineering below.

=== 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. The logarithm is one of three closely related functions. In the equation bn = x, b can be determined with radical (mathematics)s, n with logarithms, and x with exponential function. See List of logarithmic identities for several rules governing the logarithm functions.

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





















Exponential functions The exponential function ex also written as exp(x) is defined as the inverse of the natural logarithm. It is positive for every real argument x.

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

b^p = \exp({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:{| style="border:1px solid" align="center"|- align="center"! Operation with numbers !! Operation with exponents !! Logarithmic identity|- align="center"| \!\, c d || \!\, C + D || \!\, \log(c d) = \log(c) + \log(d) |- align="center"| \!\, c / d || \!\, C - D || \!\, \log(c / d) = \log(c) - \log(d) |- align="center"| \!\, c ^ d || \!\, Cd || \!\, \log(c ^ d) = d \log(c) |- align="center"| \!\, \sqrt{c} || \!\, C / d || \!\, \log(\sqrt{c}) = \frac{\log(c)}{d} |}These arithmetic properties of logarithms make such calculations much faster. The use of logarithms was an essential skill until electronic computers and calculators became available. Indeed the discovery of logarithms, just before Newton's era, had an impact in the scientific world that can be compared with that of the advent of computers in the 20th century because it made feasible many calculations that had previously been too laborious.

As an example, to approximate the product of two numbers one can look up their logarithms in a #tables of logarithms, 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 interpolation 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 marine chronometer made possible the accurate measurement of longitude 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, 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.

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 betweenthe multiplicative group of the positive real numbers and the additive group of all the reals.

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

Calculus 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: Wikisource:Table of common limits#Logarithmic and exponential functions, list of integrals of logarithmic functions.

Series for calculating the natural logarithm There are several series for calculating natural logarithms.Handbook of Mathematical Functions, National Bureau of Standards (Applied Mathematics Series no.55), June 1964, page 68. The simplest, though inefficient, is: \ln (z) = \sum_{n=1}^\infty \frac{-{(-1)}^n}{n} (z-1)^n when |z-1|> r) != 0){ r++; } return r-1; // returns -1 for x==0, floor(log2(x)) otherwise }

This algorithm can execute quickly using very few processor instructions.

Generalizations The ordinary logarithm of positive reals generalizes to negative and complex number arguments, though it is a multivalued function that needs a branch cut terminating at the branch point at 0 to make an ordinary function or principal branch. The logarithm (to base e) of a complex number z is the complex number ln(] of z, arg(z) is the Complex number#The complex plane, and i is the imaginary unit; see complex logarithm for details.

The discrete logarithm is a related notion in the theory of finite groups. It involves solving the equation bn = x, where b and x are elements of the group, and n is an integer specifying a power in the group operation. For some finite groups, it is believed that the discrete logarithm is very hard to calculate, whereas discrete exponentials are quite easy. This asymmetry has applications in public key cryptography.

The logarithm of a matrix is the inverse of the matrix exponential.

A double logarithm, \ln(\ln(x)), is the inverse function of the double exponential function. A super-logarithm or hyper-logarithm is the inverse function of the Tetration#Extension to real numbers. The super-logarithm of x grows even more slowly than the double logarithm for large x.

For each positive b not equal to 1, the function logb  (x) is an isomorphism from the group (mathematics) of positive real numbers under multiplication to the group of (all) real numbers under addition. They are the only such isomorphisms that are continuous. The logarithm function can be extended to a Haar measure in the topological group of positive real numbers under multiplication.

== History == The method of logarithms was first publicly propounded in 1614, in a book entitled Mirifici Logarithmorum Canonis Descriptio, by John Napier,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.Baron of Merchiston in Scotland (Joost Bürgi independently discovered logarithms; however, he did not publish his discovery until four years after Napier). This method contributed to the advance of science, and especially of astronomy, by making some difficult calculations possible. Prior to the advent of calculators and computers, it was used constantly in surveying, navigation, and other branches of practical mathematics. It supplanted the more involved method of prosthaphaeresis, which relied on trigonometric identities as a quick method of computing products. Besides their usefulness in computation, logarithms also fill an important place in higher theoretical mathematics.

At first, Napier called logarithms "artificial numbers" and antilogarithms "natural numbers". Later, Napier formed the word logarithm to mean a number that indicates a ratio: (logos) meaning proportion, and (arithmos) meaning number. Napier chose that because the difference of two logarithms determines the ratio of the numbers they represent, so that an arithmetic series of logarithms corresponds to a geometric series of numbers. The term antilogarithm was introduced in the late 17th century and, while never used extensively in mathematics, persisted in collections of tables until they fell into disuse.

Napier did not use a base as we now understand it, but his logarithms were, up to a scaling factor, effectively to base 1/e. For interpolation purposes and ease of calculation, it is useful to make the ratio r in the geometric series close to 1. Napier chose r = 1 - 10−7 = 0.999999 (Bürgi chose r = 1 + 10−4 = 1.0001). Napier's original logarithms did not have log 1 = 0 but rather log 107 = 0. Thus if N is a number and L is its logarithm as calculated by Napier, N = 107(1 − 10−7)L. Since (1 − 10−7)107 is approximately 1/e, this makes L/107 approximately equal to log1/e N/107.

Tables of logarithms s in the reference book Abramowitz and Stegun.Prior to the advent of computers and calculators, using logarithms meant using tables of logarithms, which had to be created manually. Base-10 logarithms are useful in computations when electronic means are not available. See common logarithm for details, including the use of characteristics and mantissas of common (i.e., base-10) logarithms.

In 1617, Henry Briggs (mathematician) published the first installment of his own table of common logarithms, containing the logarithms of all integers below 1000 to eight decimal places. This he followed, in 1624, by his Arithmetica Logarithmica, containing the logarithms of all integers from 1 to 20,000 and from 90,000 to 100,000 to fourteen places of decimals, together with a learned introduction, in which the theory and use of logarithms are fully developed. The interval from 20,000 to 90,000 was filled up by Adriaan Vlacq, a the Netherlands 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." Athenaeum, 15 June 1872. See also the Monthly Notices of the Royal Astronomical Society for May 1872. An edition of Vlacq's work, containing many corrections, was issued at Leipzig in 1794 under the title Thesaurus Logarithmorum Completus by Jurij Vega.

François Callet's seven-place table (Paris, 1795), instead of stopping at 100,000, gave the eight-place logarithms of the numbers between 100,000 and 108,000, in order to diminish the errors of interpolation, which were greatest in the early part of the table; and this addition was generally included in seven-place tables. The only important published extension of Vlacq's table was made by Mr. Sang in 1871, whose table contained the seven-place logarithms of all numbers below 200,000.

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

Besides the tables mentioned above, a great collection, called Tables du Cadastre, was constructed under the direction of Gaspard de Prony, by an original computation, under the auspices of the 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." English Cyclopaedia, Biography, Vol. IV., article "Prony." Cubic function interpolation could be used to find the logarithm of any number to a similar accuracy.

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References External links



Logarithm - Wikipedia, the free encyclopedia
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.

Common logarithm - Wikipedia, the free encyclopedia
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 log 10 (x), or sometimes Log(x) with a ...

Definition: logarithm from Online Medical Dictionary
The Online Medical Dictionary is a searchable dictionary of definitions from medicine, science and technology. ... logarithm < mathematics > One of a class of auxiliary numbers ...

logarithm definition of logarithm in the Free Online Encyclopedia.
logarithm (lŏg`ərĭ th əm) [Gr.,=relation number], number associated with a positive number, being the power to which a third number, called the base, must be raised in order to ...

Logarithm -- from Wolfram MathWorld
The logarithm log_bx for a base b and a number x is defined to be the inverse function of taking b to the power x, i.e., b^x. Therefore, for any x and b, x=log_b(b^x), (1) or ...

Natural Logarithm -- from Wolfram MathWorld
The natural logarithm lnx is the logarithm having base e, where e=2.718281828.... (1) This function can be defined lnx=int_1^x(dt)/t (2) for x>0. This definition means that e is ...

logarithm - definition of logarithm by the Free Online Dictionary ...
n. Mathematics. The power to which a base, such as 10, must be raised to produce a given number.

AskOxford: natural logarithm
natural logarithm • noun a logarithm to the base e (2.71828 &ddd;). Perform another search of the Compact Oxford English Dictionary . About this dictionary

The Natural Logarithm
The Natural Logarithm ... Definition of the Logarithm Up: The Exponential and Logarithm Previous: Differentiation Examples Contents

Napierian logarithm - Hutchinson encyclopedia article about Napierian ...
logarithm. The exponent or index of a number to a specified base - usually 10. For example, the logarithm to the base 10 of 1,000 is 3 because 10 3 = 1,000; the logarithm of 2 is 0 ...





 
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