96-528: (Redirected from IN ) Not to be confused with ln (with first letter "L"), the notation of the natural logarithm . [REDACTED] Look up IN , In , in , or in- in Wiktionary, the free dictionary. IN , In or in may refer to: Places [ edit ] India (country code IN) Indiana , United States (postal code IN) Ingolstadt , Germany (license plate code IN) In, Russia,
192-736: A 1 x d x + ∫ 1 b 1 a t a d t = ∫ 1 a 1 x d x + ∫ 1 b 1 t d t = ln a + ln b . {\displaystyle {\begin{aligned}\ln ab=\int _{1}^{ab}{\frac {1}{x}}\,dx&=\int _{1}^{a}{\frac {1}{x}}\,dx+\int _{a}^{ab}{\frac {1}{x}}\,dx\\[5pt]&=\int _{1}^{a}{\frac {1}{x}}\,dx+\int _{1}^{b}{\frac {1}{at}}a\,dt\\[5pt]&=\int _{1}^{a}{\frac {1}{x}}\,dx+\int _{1}^{b}{\frac {1}{t}}\,dt\\[5pt]&=\ln a+\ln b.\end{aligned}}} In elementary terms, this
288-460: A ) {\displaystyle \ln(a)} can easily be derived from the integral definition of ln ( a ) {\displaystyle \ln(a)} (described in the previous section) by taking the limit x → − 1 {\displaystyle x\to -1} of the integral ∫ 1 a t x d t {\displaystyle \int _{1}^{a}t^{x}dt} . By
384-418: A + d d x ln x = 1 x . {\displaystyle {\frac {d}{dx}}\ln ax={\frac {d}{dx}}(\ln a+\ln x)={\frac {d}{dx}}\ln a+{\frac {d}{dx}}\ln x={\frac {1}{x}}.} so, unlike its inverse function e a x {\displaystyle e^{ax}} , a constant in the function doesn't alter the differential. Since the natural logarithm
480-457: A is in ( 0 , 1 ) {\displaystyle (0,1)} , then the region has negative area , and the logarithm is negative. This function is a logarithm because it satisfies the fundamental multiplicative property of a logarithm: ln ( a b ) = ln a + ln b . {\displaystyle \ln(ab)=\ln a+\ln b.} This can be demonstrated by splitting
576-473: A basic tool for measurement and computation in many areas of science and engineering; in these contexts log x still often means the base ten logarithm. In mathematics log x usually means to the natural logarithm (base e ). In computer science and information theory, log often refers to binary logarithms (base 2). The following table lists common notations for logarithms to these bases. The "ISO notation" column lists designations suggested by
672-650: A family name and an element in given names "In" , an episode of Minder Imperishable Night , the eighth official game in the Japanese Touhou series Indonesian language (former ISO 639-1 language code; "id" used since November 3, 1989) In Nomine , a title given to any of numerous short pieces of English polyphonic instrumental or vocal music during the 16th and 17th centuries I.N (born Yang Jeong-in, 2001), South Korean singer and member of boy band Stray Kids See also [ edit ] Inn (disambiguation) INS Topics referred to by
768-447: A great aid to calculations before the invention of computers. Given a positive real number b such that b ≠ 1 , the logarithm of a positive real number x with respect to base b is the exponent by which b must be raised to yield x . In other words, the logarithm of x to base b is the unique real number y such that b y = x {\displaystyle b^{y}=x} . The logarithm
864-403: A positive real number a {\displaystyle a} may also be defined as the derivative of the function y = a x {\displaystyle y=a^{x}} at x = 0 {\displaystyle x=0} (assuming a x {\displaystyle a^{x}} has been previously defined without using the natural logarithm). Using
960-446: A product is the sum of the logarithms of the numbers being multiplied; the logarithm of the ratio of two numbers is the difference of the logarithms. The logarithm of the p -th power of a number is p times the logarithm of the number itself; the logarithm of a p -th root is the logarithm of the number divided by p . The following table lists these identities with examples. Each of the identities can be derived after substitution of
1056-430: A table of what in fact were effectively natural logarithms in 1619. It has been said that Speidell's logarithms were to the base e , but this is not entirely true due to complications with the values being expressed as integers . The notations ln x and log e x both refer unambiguously to the natural logarithm of x , and log x without an explicit base may also refer to the natural logarithm. This usage
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#17327795934781152-771: A town in the Jewish Autonomous Oblast Businesses and organizations [ edit ] Independent Network , a UK-based political association Indiana Northeastern Railroad (Association of American Railroads reporting mark) Indian Navy , a part of the India military Infantry , the branch of a military force that fights on foot IN Groupe , the producer of French official documents MAT Macedonian Airlines (IATA designator IN) Nam Air (IATA designator IN) Office of Intelligence and Counterintelligence , sometimes abbreviated IN Science and technology [ edit ] .in ,
1248-400: Is 1 , because e = e , while the natural logarithm of 1 is 0 , since e = 1 . The natural logarithm can be defined for any positive real number a as the area under the curve y = 1/ x from 1 to a (with the area being negative when 0 < a < 1 ). The simplicity of this definition, which is matched in many other formulas involving the natural logarithm, leads to
1344-533: Is log b y . Roughly, a continuous function is differentiable if its graph has no sharp "corners". Moreover, as the derivative of f ( x ) evaluates to ln( b ) b by the properties of the exponential function , the chain rule implies that the derivative of log b x is given by d d x log b x = 1 x ln b . {\displaystyle {\frac {d}{dx}}\log _{b}x={\frac {1}{x\ln b}}.} That is,
1440-6660: Is a graph of ln(1 + x ) and some of its Taylor polynomials around 0. These approximations converge to the function only in the region −1 < x ≤ 1 ; outside this region, the higher-degree Taylor polynomials devolve to worse approximations for the function. A useful special case for positive integers n , taking x = 1 n {\displaystyle x={\tfrac {1}{n}}} , is: ln ( n + 1 n ) = ∑ k = 1 ∞ ( − 1 ) k − 1 k n k = 1 n − 1 2 n 2 + 1 3 n 3 − 1 4 n 4 + ⋯ {\displaystyle \ln \left({\frac {n+1}{n}}\right)=\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}}{kn^{k}}}={\frac {1}{n}}-{\frac {1}{2n^{2}}}+{\frac {1}{3n^{3}}}-{\frac {1}{4n^{4}}}+\cdots } If Re ( x ) ≥ 1 / 2 , {\displaystyle \operatorname {Re} (x)\geq 1/2,} then ln ( x ) = − ln ( 1 x ) = − ∑ k = 1 ∞ ( − 1 ) k − 1 ( 1 x − 1 ) k k = ∑ k = 1 ∞ ( x − 1 ) k k x k = x − 1 x + ( x − 1 ) 2 2 x 2 + ( x − 1 ) 3 3 x 3 + ( x − 1 ) 4 4 x 4 + ⋯ {\displaystyle {\begin{aligned}\ln(x)&=-\ln \left({\frac {1}{x}}\right)=-\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}({\frac {1}{x}}-1)^{k}}{k}}=\sum _{k=1}^{\infty }{\frac {(x-1)^{k}}{kx^{k}}}\\&={\frac {x-1}{x}}+{\frac {(x-1)^{2}}{2x^{2}}}+{\frac {(x-1)^{3}}{3x^{3}}}+{\frac {(x-1)^{4}}{4x^{4}}}+\cdots \end{aligned}}} Now, taking x = n + 1 n {\displaystyle x={\tfrac {n+1}{n}}} for positive integers n , we get: ln ( n + 1 n ) = ∑ k = 1 ∞ 1 k ( n + 1 ) k = 1 n + 1 + 1 2 ( n + 1 ) 2 + 1 3 ( n + 1 ) 3 + 1 4 ( n + 1 ) 4 + ⋯ {\displaystyle \ln \left({\frac {n+1}{n}}\right)=\sum _{k=1}^{\infty }{\frac {1}{k(n+1)^{k}}}={\frac {1}{n+1}}+{\frac {1}{2(n+1)^{2}}}+{\frac {1}{3(n+1)^{3}}}+{\frac {1}{4(n+1)^{4}}}+\cdots } If Re ( x ) ≥ 0 and x ≠ 0 , {\displaystyle \operatorname {Re} (x)\geq 0{\text{ and }}x\neq 0,} then ln ( x ) = ln ( 2 x 2 ) = ln ( 1 + x − 1 x + 1 1 − x − 1 x + 1 ) = ln ( 1 + x − 1 x + 1 ) − ln ( 1 − x − 1 x + 1 ) . {\displaystyle \ln(x)=\ln \left({\frac {2x}{2}}\right)=\ln \left({\frac {1+{\frac {x-1}{x+1}}}{1-{\frac {x-1}{x+1}}}}\right)=\ln \left(1+{\frac {x-1}{x+1}}\right)-\ln \left(1-{\frac {x-1}{x+1}}\right).} Since ln ( 1 + y ) − ln ( 1 − y ) = ∑ i = 1 ∞ 1 i ( ( − 1 ) i − 1 y i − ( − 1 ) i − 1 ( − y ) i ) = ∑ i = 1 ∞ y i i ( ( − 1 ) i − 1 + 1 ) = y ∑ i = 1 ∞ y i − 1 i ( ( − 1 ) i − 1 + 1 ) = i − 1 → 2 k 2 y ∑ k = 0 ∞ y 2 k 2 k + 1 , {\displaystyle {\begin{aligned}\ln(1+y)-\ln(1-y)&=\sum _{i=1}^{\infty }{\frac {1}{i}}\left((-1)^{i-1}y^{i}-(-1)^{i-1}(-y)^{i}\right)=\sum _{i=1}^{\infty }{\frac {y^{i}}{i}}\left((-1)^{i-1}+1\right)\\&=y\sum _{i=1}^{\infty }{\frac {y^{i-1}}{i}}\left((-1)^{i-1}+1\right){\overset {i-1\to 2k}{=}}\;2y\sum _{k=0}^{\infty }{\frac {y^{2k}}{2k+1}},\end{aligned}}} we arrive at ln ( x ) = 2 ( x − 1 ) x + 1 ∑ k = 0 ∞ 1 2 k + 1 ( ( x − 1 ) 2 ( x + 1 ) 2 ) k = 2 ( x − 1 ) x + 1 ( 1 1 + 1 3 ( x − 1 ) 2 ( x + 1 ) 2 + 1 5 ( ( x − 1 ) 2 ( x + 1 ) 2 ) 2 + ⋯ ) . {\displaystyle {\begin{aligned}\ln(x)&={\frac {2(x-1)}{x+1}}\sum _{k=0}^{\infty }{\frac {1}{2k+1}}{\left({\frac {(x-1)^{2}}{(x+1)^{2}}}\right)}^{k}\\&={\frac {2(x-1)}{x+1}}\left({\frac {1}{1}}+{\frac {1}{3}}{\frac {(x-1)^{2}}{(x+1)^{2}}}+{\frac {1}{5}}{\left({\frac {(x-1)^{2}}{(x+1)^{2}}}\right)}^{2}+\cdots \right).\end{aligned}}} Using
1536-604: Is a positive real number . (If b is not a positive real number, both exponentiation and logarithm can be defined but may take several values, which makes definitions much more complicated.) One of the main historical motivations of introducing logarithms is the formula log b ( x y ) = log b x + log b y , {\displaystyle \log _{b}(xy)=\log _{b}x+\log _{b}y,} by which tables of logarithms allow multiplication and division to be reduced to addition and subtraction,
1632-466: Is as the inverse function of e x {\displaystyle e^{x}} , so that e ln ( x ) = x {\displaystyle e^{\ln(x)}=x} . Because e x {\displaystyle e^{x}} is positive and invertible for any real input x {\displaystyle x} , this definition of ln ( x ) {\displaystyle \ln(x)}
1728-426: Is called the base- b logarithm function or logarithmic function (or just logarithm ). The function log b x can also be essentially characterized by the product formula log b ( x y ) = log b x + log b y . {\displaystyle \log _{b}(xy)=\log _{b}x+\log _{b}y.} More precisely,
1824-463: Is common in mathematics, along with some scientific contexts as well as in many programming languages . In some other contexts such as chemistry , however, log x can be used to denote the common (base 10) logarithm . It may also refer to the binary (base 2) logarithm in the context of computer science , particularly in the context of time complexity . The natural logarithm can be defined in several equivalent ways. The most general definition
1920-539: Is denoted " log b x " (pronounced as "the logarithm of x to base b ", "the base- b logarithm of x ", or most commonly "the log, base b , of x "). An equivalent and more succinct definition is that the function log b is the inverse function to the function x ↦ b x {\displaystyle x\mapsto b^{x}} . Several important formulas, sometimes called logarithmic identities or logarithmic laws , relate logarithms to one another. The logarithm of
2016-518: Is denoted as log b ( x ) , or without parentheses, log b x . When the base is clear from the context or is irrelevant it is sometimes written log x . The logarithm base 10 is called the decimal or common logarithm and is commonly used in science and engineering. The natural logarithm has the number e ≈ 2.718 as its base; its use is widespread in mathematics and physics because of its very simple derivative . The binary logarithm uses base 2 and
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#17327795934782112-436: Is different from Wikidata All article disambiguation pages All disambiguation pages Natural logarithm The natural logarithm of a number is its logarithm to the base of the mathematical constant e , which is an irrational and transcendental number approximately equal to 2.718 281 828 459 . The natural logarithm of x is generally written as ln x , log e x , or sometimes, if
2208-491: Is exactly one real number x such that b x = y {\displaystyle b^{x}=y} . We let log b : R > 0 → R {\displaystyle \log _{b}\colon \mathbb {R} _{>0}\to \mathbb {R} } denote the inverse of f . That is, log b y is the unique real number x such that b x = y {\displaystyle b^{x}=y} . This function
2304-405: Is frequently used in computer science . Logarithms were introduced by John Napier in 1614 as a means of simplifying calculations. They were rapidly adopted by navigators , scientists, engineers, surveyors , and others to perform high-accuracy computations more easily. Using logarithm tables , tedious multi-digit multiplication steps can be replaced by table look-ups and simpler addition. This
2400-432: Is greater than one. In that case, log b ( x ) is an increasing function . For b < 1 , log b ( x ) tends to minus infinity instead. When x approaches zero, log b x goes to minus infinity for b > 1 (plus infinity for b < 1 , respectively). Analytic properties of functions pass to their inverses. Thus, as f ( x ) = b is a continuous and differentiable function , so
2496-575: Is possible because the logarithm of a product is the sum of the logarithms of the factors: log b ( x y ) = log b x + log b y , {\displaystyle \log _{b}(xy)=\log _{b}x+\log _{b}y,} provided that b , x and y are all positive and b ≠ 1 . The slide rule , also based on logarithms, allows quick calculations without tables, but at lower precision. The present-day notion of logarithms comes from Leonhard Euler , who connected them to
2592-403: Is related to the number of decimal digits of a positive integer x : The number of digits is the smallest integer strictly bigger than log 10 ( x ) . For example, log 10 (5986) is approximately 3.78 . The next integer above it is 4, which is the number of digits of 5986. Both the natural logarithm and the binary logarithm are used in information theory , corresponding to
2688-430: Is simply scaling by 1/ a in the horizontal direction and by a in the vertical direction. Area does not change under this transformation, but the region between a and ab is reconfigured. Because the function a /( ax ) is equal to the function 1/ x , the resulting area is precisely ln b . The number e can then be defined to be the unique real number a such that ln a = 1 . The natural logarithm of
2784-1157: Is the Taylor series for ln x {\displaystyle \ln x} around 1. A change of variables yields the Mercator series : ln ( 1 + x ) = ∑ k = 1 ∞ ( − 1 ) k − 1 k x k = x − x 2 2 + x 3 3 − ⋯ , {\displaystyle \ln(1+x)=\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}}{k}}x^{k}=x-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}-\cdots ,} valid for | x | ≤ 1 {\displaystyle |x|\leq 1} and x ≠ − 1. {\displaystyle x\neq -1.} Leonhard Euler , disregarding x ≠ − 1 {\displaystyle x\neq -1} , nevertheless applied this series to x = − 1 {\displaystyle x=-1} to show that
2880-417: Is trivially true for x ≥ 1 {\displaystyle x\geq 1} since the left hand side is negative or zero. For 0 ≤ x < 1 {\displaystyle 0\leq x<1} it is still true since both factors on the left are less than 1 (recall that α ≥ 1 {\displaystyle \alpha \geq 1} ). Thus this last statement
2976-404: Is true and by repeating our steps in reverse order we find that d d x ln ( 1 + x α ) ≤ d d x ( α x ) {\displaystyle {\frac {d}{dx}}\ln {(1+x^{\alpha })}\leq {\frac {d}{dx}}(\alpha x)} for all x {\displaystyle x} . This completes
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3072-670: Is true.) If this is true, then by multiplying the middle statement by the positive quantity ( 1 + x α ) / α {\displaystyle (1+x^{\alpha })/\alpha } and subtracting x α {\displaystyle x^{\alpha }} we would obtain x α − 1 ≤ x α + 1 {\displaystyle x^{\alpha -1}\leq x^{\alpha }+1} x α − 1 ( 1 − x ) ≤ 1 {\displaystyle x^{\alpha -1}(1-x)\leq 1} This statement
3168-1706: Is undefined at 0, ln ( x ) {\displaystyle \ln(x)} itself does not have a Maclaurin series , unlike many other elementary functions. Instead, one looks for Taylor expansions around other points. For example, if | x − 1 | ≤ 1 and x ≠ 0 , {\displaystyle \vert x-1\vert \leq 1{\text{ and }}x\neq 0,} then ln x = ∫ 1 x 1 t d t = ∫ 0 x − 1 1 1 + u d u = ∫ 0 x − 1 ( 1 − u + u 2 − u 3 + ⋯ ) d u = ( x − 1 ) − ( x − 1 ) 2 2 + ( x − 1 ) 3 3 − ( x − 1 ) 4 4 + ⋯ = ∑ k = 1 ∞ ( − 1 ) k − 1 ( x − 1 ) k k . {\displaystyle {\begin{aligned}\ln x&=\int _{1}^{x}{\frac {1}{t}}\,dt=\int _{0}^{x-1}{\frac {1}{1+u}}\,du\\&=\int _{0}^{x-1}(1-u+u^{2}-u^{3}+\cdots )\,du\\&=(x-1)-{\frac {(x-1)^{2}}{2}}+{\frac {(x-1)^{3}}{3}}-{\frac {(x-1)^{4}}{4}}+\cdots \\&=\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}(x-1)^{k}}{k}}.\end{aligned}}} This
3264-423: Is well defined for any positive x . The natural logarithm of a positive, real number a may be defined as the area under the graph of the hyperbola with equation y = 1/ x between x = 1 and x = a . This is the integral ln a = ∫ 1 a 1 x d x . {\displaystyle \ln a=\int _{1}^{a}{\frac {1}{x}}\,dx.} If
3360-411: Is written as f ( x ) = b . When b is positive and unequal to 1, we show below that f is invertible when considered as a function from the reals to the positive reals. Let b be a positive real number not equal to 1 and let f ( x ) = b . It is a standard result in real analysis that any continuous strictly monotonic function is bijective between its domain and range. This fact follows from
3456-651: The International Organization for Standardization . The history of logarithms in seventeenth-century Europe saw the discovery of a new function that extended the realm of analysis beyond the scope of algebraic methods. The method of logarithms was publicly propounded by John Napier in 1614, in a book titled Mirifici Logarithmorum Canonis Descriptio ( Description of the Wonderful Canon of Logarithms ). Prior to Napier's invention, there had been other techniques of similar scopes, such as
3552-439: The acidity of an aqueous solution . Logarithms are commonplace in scientific formulae , and in measurements of the complexity of algorithms and of geometric objects called fractals . They help to describe frequency ratios of musical intervals , appear in formulas counting prime numbers or approximating factorials , inform some models in psychophysics , and can aid in forensic accounting . The concept of logarithm as
3648-431: The decimal number system: log 10 ( 10 x ) = log 10 10 + log 10 x = 1 + log 10 x . {\displaystyle \log _{10}\,(\,10\,x\,)\ =\;\log _{10}10\ +\;\log _{10}x\ =\ 1\,+\,\log _{10}x\,.} Thus, log 10 ( x )
3744-413: The exponential function in the 18th century, and who also introduced the letter e as the base of natural logarithms. Logarithmic scales reduce wide-ranging quantities to smaller scopes. For example, the decibel (dB) is a unit used to express ratio as logarithms , mostly for signal power and amplitude (of which sound pressure is a common example). In chemistry, pH is a logarithmic measure for
3840-510: The function now known as the natural logarithm began as an attempt to perform a quadrature of a rectangular hyperbola by Grégoire de Saint-Vincent , a Belgian Jesuit residing in Prague. Archimedes had written The Quadrature of the Parabola in the third century BC, but a quadrature for the hyperbola eluded all efforts until Saint-Vincent published his results in 1647. The relation that
3936-458: The harmonic series equals the natural logarithm of 1 1 − 1 {\displaystyle {\frac {1}{1-1}}} ; that is, the logarithm of infinity. Nowadays, more formally, one can prove that the harmonic series truncated at N is close to the logarithm of N , when N is large, with the difference converging to the Euler–Mascheroni constant . The figure
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4032-433: The hyperbola with equation xy = 1 , by determination of the area of hyperbolic sectors . Their solution generated the requisite " hyperbolic logarithm " function , which had the properties now associated with the natural logarithm. An early mention of the natural logarithm was by Nicholas Mercator in his work Logarithmotechnia , published in 1668, although the mathematics teacher John Speidell had already compiled
4128-579: The intermediate value theorem . Now, f is strictly increasing (for b > 1 ), or strictly decreasing (for 0 < b < 1 ), is continuous, has domain R {\displaystyle \mathbb {R} } , and has range R > 0 {\displaystyle \mathbb {R} _{>0}} . Therefore, f is a bijection from R {\displaystyle \mathbb {R} } to R > 0 {\displaystyle \mathbb {R} _{>0}} . In other words, for each positive real number y , there
4224-421: The logarithm to base b is the inverse function of exponentiation with base b . That means that the logarithm of a number x to the base b is the exponent to which b must be raised to produce x . For example, since 1000 = 10 , the logarithm base 10 {\displaystyle 10} of 1000 is 3 , or log 10 (1000) = 3 . The logarithm of x to base b
4320-520: The prosthaphaeresis or the use of tables of progressions, extensively developed by Jost Bürgi around 1600. Napier coined the term for logarithm in Middle Latin, logarithmus , literally meaning ' ratio-number ' , derived from the Greek logos ' proportion, ratio, word ' + arithmos ' number ' . The common logarithm of a number is the index of that power of ten which equals
4416-407: The slope of the tangent touching the graph of the base- b logarithm at the point ( x , log b ( x )) equals 1/( x ln( b )) . The derivative of ln( x ) is 1/ x ; this implies that ln( x ) is the unique antiderivative of 1/ x that has the value 0 for x = 1 . It is this very simple formula that motivated to qualify as "natural" the natural logarithm; this is also one of
4512-400: The x - and the y -coordinates (or upon reflection at the diagonal line x = y ), as shown at the right: a point ( t , u = b ) on the graph of f yields a point ( u , t = log b u ) on the graph of the logarithm and vice versa. As a consequence, log b ( x ) diverges to infinity (gets bigger than any given number) if x grows to infinity, provided that b
4608-407: The 1970s, because it allows, at the expense of precision, much faster computation than techniques based on tables. A deeper study of logarithms requires the concept of a function . A function is a rule that, given one number, produces another number. An example is the function producing the x -th power of b from any real number x , where the base b is a fixed number. This function
4704-624: The above limit formula for ln ( a ) {\displaystyle \ln(a)} . The natural logarithm has the following mathematical properties: The statement is true for x = 0 {\displaystyle x=0} , and we now show that d d x ln ( 1 + x α ) ≤ d d x ( α x ) {\displaystyle {\frac {d}{dx}}\ln {(1+x^{\alpha })}\leq {\frac {d}{dx}}(\alpha x)} for all x {\displaystyle x} , which completes
4800-611: The above limit. Thus, ln ( a ) = lim n → ∞ n ( a 1 / n − 1 ) . {\displaystyle \ln(a)=\lim _{n\to \infty }n(a^{1/n}-1).} This limit formula may also be obtained by inverting the formula exp ( x ) = lim n → ∞ ( 1 + x / n ) n {\displaystyle \exp(x)=\lim _{n\to \infty }(1+x/n)^{n}} : For any positive integer n {\displaystyle n} ,
4896-407: The advance of science, especially astronomy . They were critical to advances in surveying , celestial navigation , and other domains. Pierre-Simon Laplace called logarithms As the function f ( x ) = b is the inverse function of log b x , it has been called an antilogarithm . Nowadays, this function is more commonly called an exponential function . A key tool that enabled
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#17327795934784992-461: The base e is implicit, simply log x . Parentheses are sometimes added for clarity, giving ln( x ) , log e ( x ) , or log( x ) . This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity. The natural logarithm of x is the power to which e would have to be raised to equal x . For example, ln 7.5 is 2.0149... , because e = 7.5 . The natural logarithm of e itself, ln e ,
5088-451: The base is given by: b = x 1 y , {\displaystyle b=x^{\frac {1}{y}},} which can be seen from taking the defining equation x = b log b x = b y {\displaystyle x=b^{\,\log _{b}x}=b^{y}} to the power of 1 y . {\displaystyle {\tfrac {1}{y}}.} Among all choices for
5184-405: The base, three are particularly common. These are b = 10 , b = e (the irrational mathematical constant e ≈ 2.71828183 ), and b = 2 (the binary logarithm ). In mathematical analysis , the logarithm base e is widespread because of analytical properties explained below. On the other hand, base 10 logarithms (the common logarithm ) are easy to use for manual calculations in
5280-453: The common logarithms of trigonometric functions . Another critical application was the slide rule , a pair of logarithmically divided scales used for calculation. The non-sliding logarithmic scale, Gunter's rule , was invented shortly after Napier's invention. William Oughtred enhanced it to create the slide rule—a pair of logarithmic scales movable with respect to each other. Numbers are placed on sliding scales at distances proportional to
5376-524: The definition of the derivative as a limit, this definition may be written as ln ( a ) = lim x → 0 a x − 1 x . {\displaystyle \ln(a)=\lim _{x\to 0}{\frac {a^{x}-1}{x}}.} One may rewrite this limit as an infinite sequential limit by introducing the integer variable n {\displaystyle n} and setting x = 1 / n {\displaystyle x=1/n} in
5472-1665: The derivative (for x > 0 ) can be found by using the properties of the logarithm and a definition of the exponential function. From the definition of the number e = lim u → 0 ( 1 + u ) 1 / u , {\displaystyle e=\lim _{u\to 0}(1+u)^{1/u},} the exponential function can be defined as e x = lim u → 0 ( 1 + u ) x / u = lim h → 0 ( 1 + h x ) 1 / h , {\displaystyle e^{x}=\lim _{u\to 0}(1+u)^{x/u}=\lim _{h\to 0}(1+hx)^{1/h},} where u = h x , h = u x . {\displaystyle u=hx,h={\frac {u}{x}}.} The derivative can then be found from first principles. d d x ln x = lim h → 0 ln ( x + h ) − ln x h = lim h → 0 [ 1 h ln ( x + h x ) ] = lim h → 0 [ ln ( 1 + h x ) 1 h ] all above for logarithmic properties = ln [ lim h → 0 ( 1 + h x ) 1 h ] for continuity of
5568-400: The differences between their logarithms. Sliding the upper scale appropriately amounts to mechanically adding logarithms, as illustrated here: For example, adding the distance from 1 to 2 on the lower scale to the distance from 1 to 3 on the upper scale yields a product of 6, which is read off at the lower part. The slide rule was an essential calculating tool for engineers and scientists until
5664-475: The exponent of some other quantity. For example, logarithms are used to solve for the half-life , decay constant, or unknown time in exponential decay problems. They are important in many branches of mathematics and scientific disciplines, and are used to solve problems involving compound interest . The concept of the natural logarithm was worked out by Gregoire de Saint-Vincent and Alphonse Antonio de Sarasa before 1649. Their work involved quadrature of
5760-2574: The fastest converging of the series described here. The natural logarithm can also be expressed as an infinite product: ln ( x ) = ( x − 1 ) ∏ k = 1 ∞ ( 2 1 + x 2 k ) {\displaystyle \ln(x)=(x-1)\prod _{k=1}^{\infty }\left({\frac {2}{1+{\sqrt[{2^{k}}]{x}}}}\right)} Two examples might be: ln ( 2 ) = ( 2 1 + 2 ) ( 2 1 + 2 4 ) ( 2 1 + 2 8 ) ( 2 1 + 2 16 ) . . . {\displaystyle \ln(2)=\left({\frac {2}{1+{\sqrt {2}}}}\right)\left({\frac {2}{1+{\sqrt[{4}]{2}}}}\right)\left({\frac {2}{1+{\sqrt[{8}]{2}}}}\right)\left({\frac {2}{1+{\sqrt[{16}]{2}}}}\right)...} π = ( 2 i + 2 ) ( 2 1 + i ) ( 2 1 + i 4 ) ( 2 1 + i 8 ) ( 2 1 + i 16 ) . . . {\displaystyle \pi =(2i+2)\left({\frac {2}{1+{\sqrt {i}}}}\right)\left({\frac {2}{1+{\sqrt[{4}]{i}}}}\right)\left({\frac {2}{1+{\sqrt[{8}]{i}}}}\right)\left({\frac {2}{1+{\sqrt[{16}]{i}}}}\right)...} From this identity, we can easily get that: 1 ln ( x ) = x x − 1 − ∑ k = 1 ∞ 2 − k x 2 − k 1 + x 2 − k {\displaystyle {\frac {1}{\ln(x)}}={\frac {x}{x-1}}-\sum _{k=1}^{\infty }{\frac {2^{-k}x^{2^{-k}}}{1+x^{2^{-k}}}}} For example: 1 ln ( 2 ) = 2 − 2 2 + 2 2 − 2 4 4 + 4 2 4 − 2 8 8 + 8 2 8 ⋯ {\displaystyle {\frac {1}{\ln(2)}}=2-{\frac {\sqrt {2}}{2+2{\sqrt {2}}}}-{\frac {\sqrt[{4}]{2}}{4+4{\sqrt[{4}]{2}}}}-{\frac {\sqrt[{8}]{2}}{8+8{\sqrt[{8}]{2}}}}\cdots } The natural logarithm allows simple integration of functions of
5856-433: The following formula: log b x = log k x log k b . {\displaystyle \log _{b}x={\frac {\log _{k}x}{\log _{k}b}}.} Typical scientific calculators calculate the logarithms to bases 10 and e . Logarithms with respect to any base b can be determined using either of these two logarithms by
SECTION 60
#17327795934785952-697: The form g ( x ) = f ′ ( x ) f ( x ) {\displaystyle g(x)={\frac {f'(x)}{f(x)}}} : an antiderivative of g ( x ) is given by ln ( | f ( x ) | ) {\displaystyle \ln(|f(x)|)} . This is the case because of the chain rule and the following fact: d d x ln | x | = 1 x , x ≠ 0 {\displaystyle {\frac {d}{dx}}\ln \left|x\right|={\frac {1}{x}},\ \ x\neq 0} In other words, when integrating over an interval of
6048-526: The functions f ( x ) = n ( x 1 / n − 1 ) {\displaystyle f(x)=n(x^{1/n}-1)} and g ( x ) = ( 1 + x / n ) n {\displaystyle g(x)=(1+x/n)^{n}} are easily seen to be inverses of each other, and this remains true in the limit n → ∞ {\displaystyle n\to \infty } . The above limit definition of ln (
6144-468: The identities: e ln x = x if x ∈ R + ln e x = x if x ∈ R {\displaystyle {\begin{aligned}e^{\ln x}&=x\qquad {\text{ if }}x\in \mathbb {R} _{+}\\\ln e^{x}&=x\qquad {\text{ if }}x\in \mathbb {R} \end{aligned}}} Like all logarithms,
6240-406: The integral ln x = ∫ 1 x 1 t d t , {\displaystyle \ln x=\int _{1}^{x}{\frac {1}{t}}\,dt,} then the derivative immediately follows from the first part of the fundamental theorem of calculus . On the other hand, if the natural logarithm is defined as the inverse of the (natural) exponential function, then
6336-1182: The integral of tan ( x ) {\displaystyle \tan(x)} over an interval that does not include points where tan ( x ) {\displaystyle \tan(x)} is infinite: ∫ tan x d x = ∫ sin x cos x d x = − ∫ d d x cos x cos x d x = − ln | cos x | + C = ln | sec x | + C . {\displaystyle \int \tan x\,dx=\int {\frac {\sin x}{\cos x}}\,dx=-\int {\frac {{\frac {d}{dx}}\cos x}{\cos x}}\,dx=-\ln \left|\cos x\right|+C=\ln \left|\sec x\right|+C.} The natural logarithm can be integrated using integration by parts : ∫ ln x d x = x ln x − x + C . {\displaystyle \int \ln x\,dx=x\ln x-x+C.} Logarithm In mathematics ,
6432-476: The integral that defines ln ab into two parts, and then making the variable substitution x = at (so dx = a dt ) in the second part, as follows: ln a b = ∫ 1 a b 1 x d x = ∫ 1 a 1 x d x + ∫ a a b 1 x d x = ∫ 1
6528-460: The internet top-level domain of India Inch (in), a unit of length Indium , symbol In, a chemical element Intelligent Network , a telecommunication network standard Intra-nasal ( insufflation ), a method of administrating some medications and vaccines Integrase , a retroviral enzyme Other uses [ edit ] In (album) , by the Outsiders, 1967 In (Korean name) ,
6624-482: The inverse of exponentiation extends to other mathematical structures as well. However, in general settings, the logarithm tends to be a multi-valued function. For example, the complex logarithm is the multi-valued inverse of the complex exponential function. Similarly, the discrete logarithm is the multi-valued inverse of the exponential function in finite groups; it has uses in public-key cryptography . Addition , multiplication , and exponentiation are three of
6720-462: The log base 2 ; and in photography rescaled base 2 logarithms are used to measure exposure values , light levels , exposure times , lens apertures , and film speeds in "stops". The abbreviation log x is often used when the intended base can be inferred based on the context or discipline, or when the base is indeterminate or immaterial. Common logarithms (base 10), historically used in logarithm tables and slide rules, are
6816-811: The logarithm = ln e 1 / x for the definition of e x = lim h → 0 ( 1 + h x ) 1 / h = 1 x for the definition of the ln as inverse function. {\displaystyle {\begin{aligned}{\frac {d}{dx}}\ln x&=\lim _{h\to 0}{\frac {\ln(x+h)-\ln x}{h}}\\&=\lim _{h\to 0}\left[{\frac {1}{h}}\ln \left({\frac {x+h}{x}}\right)\right]\\&=\lim _{h\to 0}\left[\ln \left(1+{\frac {h}{x}}\right)^{\frac {1}{h}}\right]\quad &&{\text{all above for logarithmic properties}}\\&=\ln \left[\lim _{h\to 0}\left(1+{\frac {h}{x}}\right)^{\frac {1}{h}}\right]\quad &&{\text{for continuity of
6912-433: The logarithm definitions x = b log b x {\displaystyle x=b^{\,\log _{b}x}} or y = b log b y {\displaystyle y=b^{\,\log _{b}y}} in the left hand sides. The logarithm log b x can be computed from the logarithms of x and b with respect to an arbitrary base k using
7008-434: The logarithm provides between a geometric progression in its argument and an arithmetic progression of values, prompted A. A. de Sarasa to make the connection of Saint-Vincent's quadrature and the tradition of logarithms in prosthaphaeresis , leading to the term "hyperbolic logarithm", a synonym for natural logarithm. Soon the new function was appreciated by Christiaan Huygens , and James Gregory . The notation Log y
7104-528: The logarithm to any base b > 1 is the only increasing function f from the positive reals to the reals satisfying f ( b ) = 1 and f ( x y ) = f ( x ) + f ( y ) . {\displaystyle f(xy)=f(x)+f(y).} As discussed above, the function log b is the inverse to the exponential function x ↦ b x {\displaystyle x\mapsto b^{x}} . Therefore, their graphs correspond to each other upon exchanging
7200-457: The logarithm}}\\&=\ln e^{1/x}\quad &&{\text{for the definition of }}e^{x}=\lim _{h\to 0}(1+hx)^{1/h}\\&={\frac {1}{x}}\quad &&{\text{for the definition of the ln as inverse function.}}\end{aligned}}} Also, we have: d d x ln a x = d d x ( ln a + ln x ) = d d x ln
7296-926: The lookups of the two logarithms, calculating their sum or difference, and looking up the antilogarithm is much faster than performing the multiplication by earlier methods such as prosthaphaeresis , which relies on trigonometric identities . Calculations of powers and roots are reduced to multiplications or divisions and lookups by c d = ( 10 log 10 c ) d = 10 d log 10 c {\displaystyle c^{d}=\left(10^{\,\log _{10}c}\right)^{d}=10^{\,d\log _{10}c}} and c d = c 1 d = 10 1 d log 10 c . {\displaystyle {\sqrt[{d}]{c}}=c^{\frac {1}{d}}=10^{{\frac {1}{d}}\log _{10}c}.} Trigonometric calculations were facilitated by tables that contained
7392-1221: The mantissa, as the characteristic can be easily determined by counting digits from the decimal point. The characteristic of 10 · x is one plus the characteristic of x , and their mantissas are the same. Thus using a three-digit log table, the logarithm of 3542 is approximated by log 10 3542 = log 10 ( 1000 ⋅ 3.542 ) = 3 + log 10 3.542 ≈ 3 + log 10 3.54 {\displaystyle {\begin{aligned}\log _{10}3542&=\log _{10}(1000\cdot 3.542)\\&=3+\log _{10}3.542\\&\approx 3+\log _{10}3.54\end{aligned}}} Greater accuracy can be obtained by interpolation : log 10 3542 ≈ 3 + log 10 3.54 + 0.2 ( log 10 3.55 − log 10 3.54 ) {\displaystyle \log _{10}3542\approx {}3+\log _{10}3.54+0.2(\log _{10}3.55-\log _{10}3.54)} The value of 10 can be determined by reverse look up in
7488-447: The most fundamental arithmetic operations. The inverse of addition is subtraction , and the inverse of multiplication is division . Similarly, a logarithm is the inverse operation of exponentiation . Exponentiation is when a number b , the base , is raised to a certain power y , the exponent , to give a value x ; this is denoted b y = x . {\displaystyle b^{y}=x.} For example, raising 2 to
7584-423: The natural logarithm maps multiplication of positive numbers into addition: ln ( x ⋅ y ) = ln x + ln y . {\displaystyle \ln(x\cdot y)=\ln x+\ln y~.} Logarithms can be defined for any positive base other than 1, not only e . However, logarithms in other bases differ only by a constant multiplier from
7680-408: The natural logarithm, and can be defined in terms of the latter, log b x = ln x / ln b = ln x ⋅ log b e {\displaystyle \log _{b}x=\ln x/\ln b=\ln x\cdot \log _{b}e} . Logarithms are useful for solving equations in which the unknown appears as
7776-425: The number. Speaking of a number as requiring so many figures is a rough allusion to common logarithm, and was referred to by Archimedes as the "order of a number". The first real logarithms were heuristic methods to turn multiplication into addition, thus facilitating rapid computation. Some of these methods used tables derived from trigonometric identities. Such methods are called prosthaphaeresis . Invention of
7872-414: The power of 3 gives 8 : 2 3 = 8. {\displaystyle 2^{3}=8.} The logarithm of base b is the inverse operation, that provides the output y from the input x . That is, y = log b x {\displaystyle y=\log _{b}x} is equivalent to x = b y {\displaystyle x=b^{y}} if b
7968-428: The power rule for antiderivatives, this integral evaluates to a x + 1 − 1 x + 1 {\displaystyle {\frac {a^{x+1}-1}{x+1}}} for all x ≠ − 1 {\displaystyle x\neq -1} . Thus, taking the limit x → − 1 {\displaystyle x\to -1} of this expression yields
8064-424: The practical use of logarithms was the table of logarithms . The first such table was compiled by Henry Briggs in 1617, immediately after Napier's invention but with the innovation of using 10 as the base. Briggs' first table contained the common logarithms of all integers in the range from 1 to 1000, with a precision of 14 digits. Subsequently, tables with increasing scope were written. These tables listed
8160-475: The previous formula: log b x = log 10 x log 10 b = log e x log e b . {\displaystyle \log _{b}x={\frac {\log _{10}x}{\log _{10}b}}={\frac {\log _{e}x}{\log _{e}b}}.} Given a number x and its logarithm y = log b x to an unknown base b ,
8256-590: The proof by the fundamental theorem of calculus . Hence, we want to show that d d x ln ( 1 + x α ) = α x α − 1 1 + x α ≤ α = d d x ( α x ) {\displaystyle {\frac {d}{dx}}\ln {(1+x^{\alpha })}={\frac {\alpha x^{\alpha -1}}{1+x^{\alpha }}}\leq \alpha ={\frac {d}{dx}}(\alpha x)} (Note that we have not yet proved that this statement
8352-464: The proof. An alternate proof is to observe that ( 1 + x α ) ≤ ( 1 + x ) α {\displaystyle (1+x^{\alpha })\leq (1+x)^{\alpha }} under the given conditions. This can be proved, e.g., by the norm inequalities. Taking logarithms and using ln ( 1 + x ) ≤ x {\displaystyle \ln(1+x)\leq x} completes
8448-411: The proof. The derivative of the natural logarithm as a real-valued function on the positive reals is given by d d x ln x = 1 x . {\displaystyle {\frac {d}{dx}}\ln x={\frac {1}{x}}.} How to establish this derivative of the natural logarithm depends on how it is defined firsthand. If the natural logarithm is defined as
8544-495: The real line that does not include x = 0 {\displaystyle x=0} , then ∫ 1 x d x = ln | x | + C {\displaystyle \int {\frac {1}{x}}\,dx=\ln |x|+C} where C is an arbitrary constant of integration . Likewise, when the integral is over an interval where f ( x ) ≠ 0 {\displaystyle f(x)\neq 0} , For example, consider
8640-1046: The same table, since the logarithm is a monotonic function . The product and quotient of two positive numbers c and d were routinely calculated as the sum and difference of their logarithms. The product cd or quotient c / d came from looking up the antilogarithm of the sum or difference, via the same table: c d = 10 log 10 c 10 log 10 d = 10 log 10 c + log 10 d {\displaystyle cd=10^{\,\log _{10}c}\,10^{\,\log _{10}d}=10^{\,\log _{10}c\,+\,\log _{10}d}} and c d = c d − 1 = 10 log 10 c − log 10 d . {\displaystyle {\frac {c}{d}}=cd^{-1}=10^{\,\log _{10}c\,-\,\log _{10}d}.} For manual calculations that demand any appreciable precision, performing
8736-401: The same term [REDACTED] This disambiguation page lists articles associated with the title In . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=In&oldid=1259767859 " Category : Disambiguation pages Hidden categories: Short description
8832-1000: The substitution x = n + 1 n {\displaystyle x={\tfrac {n+1}{n}}} again for positive integers n , we get: ln ( n + 1 n ) = 2 2 n + 1 ∑ k = 0 ∞ 1 ( 2 k + 1 ) ( ( 2 n + 1 ) 2 ) k = 2 ( 1 2 n + 1 + 1 3 ( 2 n + 1 ) 3 + 1 5 ( 2 n + 1 ) 5 + ⋯ ) . {\displaystyle {\begin{aligned}\ln \left({\frac {n+1}{n}}\right)&={\frac {2}{2n+1}}\sum _{k=0}^{\infty }{\frac {1}{(2k+1)((2n+1)^{2})^{k}}}\\&=2\left({\frac {1}{2n+1}}+{\frac {1}{3(2n+1)^{3}}}+{\frac {1}{5(2n+1)^{5}}}+\cdots \right).\end{aligned}}} This is, by far,
8928-423: The term "natural". The definition of the natural logarithm can then be extended to give logarithm values for negative numbers and for all non-zero complex numbers , although this leads to a multi-valued function : see complex logarithm for more. The natural logarithm function, if considered as a real-valued function of a positive real variable, is the inverse function of the exponential function , leading to
9024-488: The use of nats or bits as the fundamental units of information, respectively. Binary logarithms are also used in computer science , where the binary system is ubiquitous; in music theory , where a pitch ratio of two (the octave ) is ubiquitous and the number of cents between any two pitches is a scaled version of the binary logarithm, or log 2 times 1200, of the pitch ratio (that is, 100 cents per semitone in conventional equal temperament ), or equivalently
9120-455: The values of log 10 x for any number x in a certain range, at a certain precision. Base-10 logarithms were universally used for computation, hence the name common logarithm, since numbers that differ by factors of 10 have logarithms that differ by integers. The common logarithm of x can be separated into an integer part and a fractional part , known as the characteristic and mantissa . Tables of logarithms need only include
9216-673: Was adopted by Leibniz in 1675, and the next year he connected it to the integral ∫ d y y . {\textstyle \int {\frac {dy}{y}}.} Before Euler developed his modern conception of complex natural logarithms, Roger Cotes had a nearly equivalent result when he showed in 1714 that log ( cos θ + i sin θ ) = i θ . {\displaystyle \log(\cos \theta +i\sin \theta )=i\theta .} By simplifying difficult calculations before calculators and computers became available, logarithms contributed to
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