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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 2.718281828459. The natural logarithm of x is generally written as ln x, loge x, or sometimes, if the base e is implicit, simply log x.[1][2] Parentheses are sometimes added for clarity, giving ln(x), loge(x), or log(x). This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity.

Natural logarithm
Graph of part of the natural logarithm function. The function slowly grows to positive infinity as x increases, and slowly goes to negative infinity as x approaches 0 ("slowly" as compared to any power law of x).
General information
General definition
Motivation of inventionAnalytic proofs
Fields of applicationPure and applied mathematics
Domain, Codomain and Image
Domain
Codomain
Image
Specific values
Value at +∞+∞
Value at e1
Specific features
Asymptote
Root1
Inverse
Derivative
Antiderivative

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 e2.0149... = 7.5. The natural logarithm of e itself, ln e, is 1, because e1 = e, while the natural logarithm of 1 is 0, since e0 = 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[3] (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 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 the identities:

Like all logarithms, the natural logarithm maps multiplication of positive numbers into addition:

[4]

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 the natural logarithm, and can be defined in terms of the latter, .

Logarithms are useful for solving equations in which the unknown appears as 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.

History

The concept of the natural logarithm was worked out by Gregoire de Saint-Vincent and Alphonse Antonio de Sarasa before 1649.[5] Their work involved quadrature of 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,[6] although the mathematics teacher John Speidell had already compiled a table of what in fact were effectively natural logarithms in 1619.[7] 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.[7]: 152 

Notational conventions

The notations ln x and loge 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 is common in mathematics, along with some scientific contexts as well as in many programming languages.[nb 1] 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.

Definitions

The natural logarithm can be defined in several equivalent ways.

Inverse of exponential

The most general definition is as the inverse function of  , so that  . Because   is positive and invertible for any real input  , this definition of   is well defined for any positive x. For the complex numbers,   is not invertible, so   is a multivalued function. In order to make   a proper, single-output function, we therefore need to restrict it to a particular principal branch, often denoted by  . As the inverse function of  ,   can be defined by inverting the usual definition of  :

 

Doing so yields:

 

This definition therefore derives its own principal branch from the principal branch of nth roots.

Integral definition

 
ln a as the area of the shaded region under the curve f(x) = 1/x from 1 to a. If a is less than 1, the area taken to be negative.
 
The area under the hyperbola satisfies the logarithm rule. Here A(s,t) denotes the area under the hyperbola between s and t.

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[3]

 

If a is less than 1, then this area is considered to be negative.

This function is a logarithm because it satisfies the fundamental multiplicative property of a logarithm:[4]

 

This can be demonstrated by splitting 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:

 

In elementary terms, this 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 also has an improper integral representation,[8] which can be derived with Fubini's theorem as follows:

 

Properties

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Proof

The statement is true for  , and we now show that   for all  , which completes the proof by the fundamental theorem of calculus. Hence, we want to show that

 

(Note that we have not yet proved that this statement is true.) If this is true, then by multiplying the middle statement by the positive quantity   and subtracting   we would obtain

 
 

This statement is trivially true for   since the left hand side is negative or zero. For   it is still true since both factors on the left are less than 1 (recall that  ). Thus this last statement is true and by repeating our steps in reverse order we find that   for all  . This completes the proof.

An alternate proof is to observe that   under the given conditions. This can be proved, e.g., by the norm inequalities. Taking logarithms and using   completes the proof.

Derivative

The derivative of the natural logarithm as a real-valued function on the positive reals is given by[3]

 

How to establish this derivative of the natural logarithm depends on how it is defined firsthand. If the natural logarithm is defined as the integral

 

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 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   the exponential function can be defined as  , where   The derivative can then be found from first principles.

 

Also, we have:

 

so, unlike its inverse function  , a constant in the function doesn't alter the differential.

Series

 
The Taylor polynomials for ln(1 + x) only provide accurate approximations in the range −1 < x ≤ 1. Beyond some x > 1, the Taylor polynomials of higher degree are increasingly worse approximations.

Since the natural logarithm is undefined at 0,   itself does not have a Maclaurin series, unlike many other elementary functions. Instead, one looks for Taylor expansions around other points. For example, if   then[9]

 

This is the Taylor series for ln x around 1. A change of variables yields the Mercator series:

 

valid for |x| ≤ 1 and x ≠ −1.

Leonhard Euler,[10] disregarding  , nevertheless applied this series to x = −1, in order to show that the harmonic series equals the (natural) logarithm of 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.

At right is a picture 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 of this region the higher-degree Taylor polynomials evolve to worse approximations for the function.

A useful special case for positive integers n, taking  , is:

 

If   then

 

Now, taking   for positive integers n, we get:

 

If   then

 

Since

 

we arrive at

 

Using the substitution   again for positive integers n, we get:

 

This is, by far, the fastest converging of the series described here.

The natural logarithm can also be expressed as an infinite product:[11]

 

Two examples might be:

 
 

From this identity, we can easily get that:

 

For example:

 

The natural logarithm in integration

The natural logarithm allows simple integration of functions of the form g(x) = f '(x)/f(x): an antiderivative of g(x) is given by ln(|f(x)|). This is the case because of the chain rule and the following fact:

 

In other words, if   is a real number with  , then

 [12]

and

 

Here is an example in the case of g(x) = tan(x):

 

Letting f(x) = cos(x):

 
 

where C is an arbitrary constant of integration.

The natural logarithm can be integrated using integration by parts:

 

Let:

 
 

then:

 

Efficient computation

For ln(x) where x > 1, the closer the value of x is to 1, the faster the rate of convergence of its Taylor series centered at 1. The identities associated with the logarithm can be leveraged to exploit this:

 

Such techniques were used before calculators, by referring to numerical tables and performing manipulations such as those above.

Natural logarithm of 10

The natural logarithm of 10, which has the decimal expansion 2.30258509...,[13] plays a role for example in the computation of natural logarithms of numbers represented in scientific notation, as a mantissa multiplied by a power of 10:

 

This means that one can effectively calculate the logarithms of numbers with very large or very small magnitude using the logarithms of a relatively small set of decimals in the range [1, 10).

High precision

To compute the natural logarithm with many digits of precision, the Taylor series approach is not efficient since the convergence is slow. Especially if x is near 1, a good alternative is to use Halley's method or Newton's method to invert the exponential function, because the series of the exponential function converges more quickly. For finding the value of y to give exp(y) − x = 0 using Halley's method, or equivalently to give exp(y/2) − x exp(−y/2) = 0 using Newton's method, the iteration simplifies to

 

which has cubic convergence to ln(x).

Another alternative for extremely high precision calculation is the formula[14][15]

 

where M denotes the arithmetic-geometric mean of 1 and 4/s, and

 

with m chosen so that p bits of precision is attained. (For most purposes, the value of 8 for m is sufficient.) In fact, if this method is used, Newton inversion of the natural logarithm may conversely be used to calculate the exponential function efficiently. (The constants ln 2 and π can be pre-computed to the desired precision using any of several known quickly converging series.) Or, the following formula can be used:

 

where

 

are the Jacobi theta functions.[16]

Based on a proposal by William Kahan and first implemented in the Hewlett-Packard HP-41C calculator in 1979 (referred to under "LN1" in the display, only), some calculators, operating systems (for example Berkeley UNIX 4.3BSD[17]), computer algebra systems and programming languages (for example C99[18]) provide a special natural logarithm plus 1 function, alternatively named LNP1,[19][20] or log1p[18] to give more accurate results for logarithms close to zero by passing arguments x, also close to zero, to a function log1p(x), which returns the value ln(1+x), instead of passing a value y close to 1 to a function returning ln(y).[18][19][20] The function log1p avoids in the floating point arithmetic a near cancelling of the absolute term 1 with the second term from the Taylor expansion of the ln. This keeps the argument, the result, and intermediate steps all close to zero where they can be most accurately represented as floating-point numbers.[19][20]

In addition to base e the IEEE 754-2008 standard defines similar logarithmic functions near 1 for binary and decimal logarithms: log2(1 + x) and log10(1 + x).

Similar inverse functions named "expm1",[18] "expm"[19][20] or "exp1m" exist as well, all with the meaning of expm1(x) = exp(x) − 1.[nb 2]

An identity in terms of the inverse hyperbolic tangent,

 

gives a high precision value for small values of x on systems that do not implement log1p(x).

Computational complexity

The computational complexity of computing the natural logarithm using the arithmetic-geometric mean (for both of the above methods) is O(M(n) ln n). Here n is the number of digits of precision at which the natural logarithm is to be evaluated and M(n) is the computational complexity of multiplying two n-digit numbers.

Continued fractions

While no simple continued fractions are available, several generalized continued fractions are, including:

 
 

These continued fractions—particularly the last—converge rapidly for values close to 1. However, the natural logarithms of much larger numbers can easily be computed, by repeatedly adding those of smaller numbers, with similarly rapid convergence.

For example, since 2 = 1.253 × 1.024, the natural logarithm of 2 can be computed as:

 

Furthermore, since 10 = 1.2510 × 1.0243, even the natural logarithm of 10 can be computed similarly as:

 

The reciprocal of the natural logarithm can be also written in this way:

 

For example:

 

Complex logarithms

The exponential function can be extended to a function which gives a complex number as ez for any arbitrary complex number z; simply use the infinite series with x=z complex. This exponential function can be inverted to form a complex logarithm that exhibits most of the properties of the ordinary logarithm. There are two difficulties involved: no x has ex = 0; and it turns out that e2 = 1 = e0. Since the multiplicative property still works for the complex exponential function, ez = ez+2kiπ, for all complex z and integers k.

So the logarithm cannot be defined for the whole complex plane, and even then it is multi-valued—any complex logarithm can be changed into an "equivalent" logarithm by adding any integer multiple of 2 at will. The complex logarithm can only be single-valued on the cut plane. For example, ln i = /2 or 5/2 or -3/2, etc.; and although i4 = 1, 4 ln i can be defined as 2, or 10 or −6, and so on.

See also

Notes

  1. ^ Including C, C++, SAS, MATLAB, Mathematica, Fortran, and some BASIC dialects
  2. ^ For a similar approach to reduce round-off errors of calculations for certain input values see trigonometric functions like versine, vercosine, coversine, covercosine, haversine, havercosine, hacoversine, hacovercosine, exsecant and excosecant.

References

  1. ^ G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers, 4th Ed., Oxford 1975, footnote to paragraph 1.7: "log x is, of course, the 'Naperian' logarithm of x, to base e. 'Common' logarithms have no mathematical interest".
  2. ^ Mortimer, Robert G. (2005). Mathematics for physical chemistry (3rd ed.). Academic Press. p. 9. ISBN 0-12-508347-5. Extract of page 9
  3. ^ a b c Weisstein, Eric W. "Natural Logarithm". mathworld.wolfram.com. Retrieved 2020-08-29.
  4. ^ a b "Rules, Examples, & Formulas". Logarithm. Encyclopedia Britannica. Retrieved 2020-08-29.
  5. ^ Burn, R.P. (2001). Alphonse Antonio de Sarasa and Logarithms. Historia Mathematica. pp. 28:1–17.
  6. ^ O'Connor, J. J.; Robertson, E. F. (September 2001). "The number e". The MacTutor History of Mathematics archive. Retrieved 2009-02-02.
  7. ^ a b Cajori, Florian (1991). A History of Mathematics (5th ed.). AMS Bookstore. p. 152. ISBN 0-8218-2102-4.
  8. ^ An improper integral representation of the natural logarithm., retrieved 2022-09-24
  9. ^ ""Logarithmic Expansions" at Math2.org".
  10. ^ Leonhard Euler, Introductio in Analysin Infinitorum. Tomus Primus. Bousquet, Lausanne 1748. Exemplum 1, p. 228; quoque in: Opera Omnia, Series Prima, Opera Mathematica, Volumen Octavum, Teubner 1922
  11. ^ RUFFA, Anthony. "A PROCEDURE FOR GENERATING INFINITE SERIES IDENTITIES" (PDF). International Journal of Mathematics and Mathematical Sciences. International Journal of Mathematics and Mathematical Sciences. Retrieved 2022-02-27. (Page 3654, equation 2.6)
  12. ^ For a detailed proof see for instance: George B. Thomas, Jr and Ross L. Finney, Calculus and Analytic Geometry, 5th edition, Addison-Wesley 1979, Section 6-5 pages 305-306.
  13. ^ OEISA002392
  14. ^ Sasaki, T.; Kanada, Y. (1982). "Practically fast multiple-precision evaluation of log(x)". Journal of Information Processing. 5 (4): 247–250. Retrieved 2011-03-30.
  15. ^ Ahrendt, Timm (1999). "Fast Computations of the Exponential Function". Stacs 99. Lecture Notes in Computer Science. 1564: 302–312. doi:10.1007/3-540-49116-3_28. ISBN 978-3-540-65691-3.
  16. ^ Borwein, Jonathan M.; Borwein, Peter B. (1987). Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (First ed.). Wiley-Interscience. ISBN 0-471-83138-7. page 225
  17. ^ Beebe, Nelson H. F. (2017-08-22). "Chapter 10.4. Logarithm near one". The Mathematical-Function Computation Handbook - Programming Using the MathCW Portable Software Library (1 ed.). Salt Lake City, UT, USA: Springer International Publishing AG. pp. 290–292. doi:10.1007/978-3-319-64110-2. ISBN 978-3-319-64109-6. LCCN 2017947446. S2CID 30244721. In 1987, Berkeley UNIX 4.3BSD introduced the log1p() function
  18. ^ a b c d Beebe, Nelson H. F. (2002-07-09). "Computation of expm1 = exp(x)−1" (PDF). 1.00. Salt Lake City, Utah, USA: Department of Mathematics, Center for Scientific Computing, University of Utah. Retrieved 2015-11-02.
  19. ^ a b c d HP 48G Series – Advanced User's Reference Manual (AUR) (4 ed.). Hewlett-Packard. December 1994 [1993]. HP 00048-90136, 0-88698-01574-2. Retrieved 2015-09-06.
  20. ^ a b c d HP 50g / 49g+ / 48gII graphing calculator advanced user's reference manual (AUR) (2 ed.). Hewlett-Packard. 2009-07-14 [2005]. HP F2228-90010. Retrieved 2015-10-10. Searchable PDF

natural, logarithm, base, redirects, here, numbering, system, which, uses, base, integer, base, numeration, base, natural, logarithm, number, logarithm, base, mathematical, constant, which, irrational, transcendental, number, approximately, equal, 718281, natu. Base e redirects here For the numbering system which uses e as its base see Non integer base of numeration Base e 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 718281 828 459 The natural logarithm of x is generally written as ln x loge x or sometimes if the base e is implicit simply log x 1 2 Parentheses are sometimes added for clarity giving ln x loge x or log x This is done particularly when the argument to the logarithm is not a single symbol so as to prevent ambiguity Natural logarithmGraph of part of the natural logarithm function The function slowly grows to positive infinity as x increases and slowly goes to negative infinity as x approaches 0 slowly as compared to any power law of x General informationGeneral definitionln x log e x displaystyle ln x log e x Motivation of inventionAnalytic proofsFields of applicationPure and applied mathematicsDomain Codomain and ImageDomainR gt 0 displaystyle mathbb R gt 0 CodomainR displaystyle mathbb R ImageR displaystyle mathbb R Specific valuesValue at Value at e1Specific featuresAsymptotex 0 displaystyle x 0 Root1Inverseexp x displaystyle exp x Derivatived d x ln x 1 x x gt 0 displaystyle dfrac d dx ln x dfrac 1 x x gt 0 Antiderivative ln x d x x ln x 1 C displaystyle int ln x dx x left ln x 1 right C 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 e2 0149 7 5 The natural logarithm of e itself ln e is 1 because e1 e while the natural logarithm of 1 is 0 since e0 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 3 with the area being negative when 0 lt a lt 1 The simplicity of this definition which is matched in many other formulas involving the natural logarithm leads to 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 the identities e ln x x if x is strictly positive ln e x x if x is any real number displaystyle begin aligned e ln x amp x qquad text if x text is strictly positive ln e x amp x qquad text if x text is any real number end aligned Like all logarithms 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 4 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 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 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 Contents 1 History 2 Notational conventions 3 Definitions 3 1 Inverse of exponential 3 2 Integral definition 4 Properties 5 Derivative 6 Series 7 The natural logarithm in integration 8 Efficient computation 8 1 Natural logarithm of 10 8 2 High precision 8 3 Computational complexity 9 Continued fractions 10 Complex logarithms 11 See also 12 Notes 13 ReferencesHistory EditMain article History of logarithms The concept of the natural logarithm was worked out by Gregoire de Saint Vincent and Alphonse Antonio de Sarasa before 1649 5 Their work involved quadrature of 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 6 although the mathematics teacher John Speidell had already compiled a table of what in fact were effectively natural logarithms in 1619 7 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 7 152 Notational conventions EditThe notations ln x and loge 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 is common in mathematics along with some scientific contexts as well as in many programming languages nb 1 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 Definitions EditThe natural logarithm can be defined in several equivalent ways Inverse of exponential Edit The most general definition 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 is well defined for any positive x For the complex numbers e z displaystyle e z is not invertible so ln z displaystyle ln z is a multivalued function In order to make ln z displaystyle ln z a proper single output function we therefore need to restrict it to a particular principal branch often denoted by Ln z displaystyle operatorname Ln z As the inverse function of e z displaystyle e z ln z displaystyle ln z can be defined by inverting the usual definition of e z displaystyle e z e z lim n 1 z n n displaystyle e z lim n to infty left 1 frac z n right n Doing so yields ln z lim n n z n 1 displaystyle ln z lim n to infty n cdot sqrt n z 1 This definition therefore derives its own principal branch from the principal branch of nth roots Integral definition Edit ln a as the area of the shaded region under the curve f x 1 x from 1 to a If a is less than 1 the area taken to be negative The area under the hyperbola satisfies the logarithm rule Here A s t denotes the area under the hyperbola between s and t 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 3 ln a 1 a 1 x d x displaystyle ln a int 1 a frac 1 x dx If a is less than 1 then this area is considered to be negative This function is a logarithm because it satisfies the fundamental multiplicative property of a logarithm 4 ln a b ln a ln b displaystyle ln ab ln a ln b This can be demonstrated by splitting 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 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 amp int 1 a frac 1 x dx int a ab frac 1 x dx 5pt amp int 1 a frac 1 x dx int 1 b frac 1 at a dt 5pt amp int 1 a frac 1 x dx int 1 b frac 1 t dt 5pt amp ln a ln b end aligned In elementary terms this 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 also has an improper integral representation 8 which can be derived with Fubini s theorem as follows ln x 1 x 1 u d u 1 x 0 e t u d t d u 0 1 x e t u d u d t 0 e t e t x t d t displaystyle ln left x right int 1 x frac 1 u du int 1 x int 0 infty e tu dt du int 0 infty int 1 x e tu du dt int 0 infty frac e t e tx t dt Properties Editln 1 0 displaystyle ln 1 0 ln e 1 displaystyle ln e 1 ln x y ln x ln y for x gt 0 and y gt 0 displaystyle ln xy ln x ln y quad text for x gt 0 text and y gt 0 ln x y ln x ln y displaystyle ln x y ln x ln y ln x y y ln x for x gt 0 displaystyle ln x y y ln x quad text for x gt 0 ln x lt ln y for 0 lt x lt y displaystyle ln x lt ln y quad text for 0 lt x lt y lim x 0 ln 1 x x 1 displaystyle lim x to 0 frac ln 1 x x 1 lim a 0 x a 1 a ln x for x gt 0 displaystyle lim alpha to 0 frac x alpha 1 alpha ln x quad text for x gt 0 x 1 x ln x x 1 for x gt 0 displaystyle frac x 1 x leq ln x leq x 1 quad text for quad x gt 0 ln 1 x a a x for x 0 and a 1 displaystyle ln 1 x alpha leq alpha x quad text for quad x geq 0 text and alpha geq 1 ProofThe statement is true for x 0 displaystyle x 0 and we now show that d d x ln 1 x a d d x a x displaystyle frac d dx ln 1 x alpha leq frac d dx alpha x for all x displaystyle x which completes the proof by the fundamental theorem of calculus Hence we want to show that d d x ln 1 x a a x a 1 1 x a a d d x a 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 is true If this is true then by multiplying the middle statement by the positive quantity 1 x a a displaystyle 1 x alpha alpha and subtracting x a displaystyle x alpha we would obtain x a 1 x a 1 displaystyle x alpha 1 leq x alpha 1 x a 1 1 x 1 displaystyle x alpha 1 1 x leq 1 This statement is trivially true for x 1 displaystyle x geq 1 since the left hand side is negative or zero For 0 x lt 1 displaystyle 0 leq x lt 1 it is still true since both factors on the left are less than 1 recall that a 1 displaystyle alpha geq 1 Thus this last statement is true and by repeating our steps in reverse order we find that d d x ln 1 x a d d x a x displaystyle frac d dx ln 1 x alpha leq frac d dx alpha x for all x displaystyle x This completes the proof An alternate proof is to observe that 1 x a 1 x a 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 the proof Derivative EditThe derivative of the natural logarithm as a real valued function on the positive reals is given by 3 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 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 the derivative for x gt 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 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 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 amp lim h to 0 frac ln x h ln x h amp lim h to 0 left frac 1 h ln left frac x h x right right amp lim h to 0 left ln left left 1 frac h x right frac 1 h right right quad amp amp text all above for logarithmic properties amp ln left lim h to 0 left 1 frac h x right frac 1 h right quad amp amp text for continuity of the logarithm amp ln e 1 x quad amp amp text for the definition of e x lim h to 0 1 hx 1 h amp frac 1 x quad amp amp 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 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 Series Edit The Taylor polynomials for ln 1 x only provide accurate approximations in the range 1 lt x 1 Beyond some x gt 1 the Taylor polynomials of higher degree are increasingly worse approximations Since the natural logarithm 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 9 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 amp int 1 x frac 1 t dt int 0 x 1 frac 1 1 u du amp int 0 x 1 1 u u 2 u 3 cdots du amp x 1 frac x 1 2 2 frac x 1 3 3 frac x 1 4 4 cdots amp sum k 1 infty frac 1 k 1 x 1 k k end aligned This is the Taylor series for 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 and x 1 Leonhard Euler 10 disregarding x 1 displaystyle x neq 1 nevertheless applied this series to x 1 in order to show that the harmonic series equals the natural logarithm of 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 At right is a picture of ln 1 x and some of its Taylor polynomials around 0 These approximations converge to the function only in the region 1 lt x 1 outside of this region the higher degree Taylor polynomials evolve 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 amp 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 amp 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 amp 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 amp 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 amp frac 2 x 1 x 1 sum k 0 infty frac 1 2k 1 left frac x 1 2 x 1 2 right k amp 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 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 amp frac 2 2n 1 sum k 0 infty frac 1 2k 1 2n 1 2 k amp 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 the fastest converging of the series described here The natural logarithm can also be expressed as an infinite product 11 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 p 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 in integration EditThe natural logarithm allows simple integration of functions of the form g x f x f x an antiderivative of g x is given by ln f x This is the case because of the chain rule and the following fact d d x ln x 1 x displaystyle frac d dx ln left x right frac 1 x In other words if x displaystyle x is a real number with x 0 displaystyle x not 0 then 1 x d x ln x C displaystyle int frac 1 x dx ln x C 12 and f x f x d x ln f x C displaystyle int frac f x f x dx ln f x C Here is an example in the case of g x tan x tan x d x sin x cos x d x tan x d x d d x cos x cos x d x displaystyle begin aligned amp int tan x dx int frac sin x cos x dx 6pt amp int tan x dx int frac frac d dx cos x cos x dx end aligned Letting f x cos x tan x d x ln cos x C displaystyle int tan x dx ln left cos x right C tan x d x ln sec x C displaystyle int tan x dx ln left sec x right C where C is an arbitrary constant of integration 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 Let u ln x d u d x x displaystyle u ln x Rightarrow du frac dx x d v d x v x displaystyle dv dx Rightarrow v x then ln x d x x ln x x x d x x ln x 1 d x x ln x x C displaystyle begin aligned int ln x dx amp x ln x int frac x x dx amp x ln x int 1 dx amp x ln x x C end aligned Efficient computation EditFor ln x where x gt 1 the closer the value of x is to 1 the faster the rate of convergence of its Taylor series centered at 1 The identities associated with the logarithm can be leveraged to exploit this ln 123 456 ln 1 23456 10 2 ln 1 23456 ln 10 2 ln 1 23456 2 ln 10 ln 1 23456 2 2 3025851 displaystyle begin aligned ln 123 456 amp ln 1 23456 cdot 10 2 amp ln 1 23456 ln 10 2 amp ln 1 23456 2 ln 10 amp approx ln 1 23456 2 cdot 2 3025851 end aligned Such techniques were used before calculators by referring to numerical tables and performing manipulations such as those above Natural logarithm of 10 Edit The natural logarithm of 10 which has the decimal expansion 2 30258509 13 plays a role for example in the computation of natural logarithms of numbers represented in scientific notation as a mantissa multiplied by a power of 10 ln a 10 n ln a n ln 10 displaystyle ln a cdot 10 n ln a n ln 10 This means that one can effectively calculate the logarithms of numbers with very large or very small magnitude using the logarithms of a relatively small set of decimals in the range 1 10 High precision Edit To compute the natural logarithm with many digits of precision the Taylor series approach is not efficient since the convergence is slow Especially if x is near 1 a good alternative is to use Halley s method or Newton s method to invert the exponential function because the series of the exponential function converges more quickly For finding the value of y to give exp y x 0 using Halley s method or equivalently to give exp y 2 x exp y 2 0 using Newton s method the iteration simplifies to y n 1 y n 2 x exp y n x exp y n displaystyle y n 1 y n 2 cdot frac x exp y n x exp y n which has cubic convergence to ln x Another alternative for extremely high precision calculation is the formula 14 15 ln x p 2 M 1 4 s m ln 2 displaystyle ln x approx frac pi 2M 1 4 s m ln 2 where M denotes the arithmetic geometric mean of 1 and 4 s and s x 2 m gt 2 p 2 displaystyle s x2 m gt 2 p 2 with m chosen so that p bits of precision is attained For most purposes the value of 8 for m is sufficient In fact if this method is used Newton inversion of the natural logarithm may conversely be used to calculate the exponential function efficiently The constants ln 2 and p can be pre computed to the desired precision using any of several known quickly converging series Or the following formula can be used ln x p M 8 2 2 1 x 8 3 2 1 x x 1 displaystyle ln x frac pi M left theta 2 2 1 x theta 3 2 1 x right quad x in 1 infty where 8 2 x n Z x n 1 2 2 8 3 x n Z x n 2 displaystyle theta 2 x sum n in mathbb Z x n 1 2 2 quad theta 3 x sum n in mathbb Z x n 2 are the Jacobi theta functions 16 Based on a proposal by William Kahan and first implemented in the Hewlett Packard HP 41C calculator in 1979 referred to under LN1 in the display only some calculators operating systems for example Berkeley UNIX 4 3BSD 17 computer algebra systems and programming languages for example C99 18 provide a special natural logarithm plus 1 function alternatively named LNP1 19 20 or log1p 18 to give more accurate results for logarithms close to zero by passing arguments x also close to zero to a function log1p x which returns the value ln 1 x instead of passing a value y close to 1 to a function returning ln y 18 19 20 The function log1p avoids in the floating point arithmetic a near cancelling of the absolute term 1 with the second term from the Taylor expansion of the ln This keeps the argument the result and intermediate steps all close to zero where they can be most accurately represented as floating point numbers 19 20 In addition to base e the IEEE 754 2008 standard defines similar logarithmic functions near 1 for binary and decimal logarithms log2 1 x and log10 1 x Similar inverse functions named expm1 18 expm 19 20 or exp1m exist as well all with the meaning of expm1 x exp x 1 nb 2 An identity in terms of the inverse hyperbolic tangent l o g 1 p x log 1 x 2 a r t a n h x 2 x displaystyle mathrm log1p x log 1 x 2 mathrm artanh left frac x 2 x right gives a high precision value for small values of x on systems that do not implement log1p x Computational complexity Edit Main article Computational complexity of mathematical operations The computational complexity of computing the natural logarithm using the arithmetic geometric mean for both of the above methods is O M n ln n Here n is the number of digits of precision at which the natural logarithm is to be evaluated and M n is the computational complexity of multiplying two n digit numbers Continued fractions EditWhile no simple continued fractions are available several generalized continued fractions are including ln 1 x x 1 1 x 2 2 x 3 3 x 4 4 x 5 5 x 1 0 x 1 2 x 2 1 x 2 2 x 3 2 x 3 2 x 4 3 x 4 2 x 5 4 x displaystyle begin aligned ln 1 x amp frac x 1 1 frac x 2 2 frac x 3 3 frac x 4 4 frac x 5 5 cdots 5pt amp cfrac x 1 0x cfrac 1 2 x 2 1x cfrac 2 2 x 3 2x cfrac 3 2 x 4 3x cfrac 4 2 x 5 4x ddots end aligned ln 1 x y x y 1 x 2 1 x 3 y 2 x 2 2 x 5 y 3 x 2 2 x 2 y x 1 x 2 3 2 y x 2 x 2 5 2 y x 3 x 2 7 2 y x displaystyle begin aligned ln left 1 frac x y right amp cfrac x y cfrac 1x 2 cfrac 1x 3y cfrac 2x 2 cfrac 2x 5y cfrac 3x 2 ddots 5pt amp cfrac 2x 2y x cfrac 1x 2 3 2y x cfrac 2x 2 5 2y x cfrac 3x 2 7 2y x ddots end aligned These continued fractions particularly the last converge rapidly for values close to 1 However the natural logarithms of much larger numbers can easily be computed by repeatedly adding those of smaller numbers with similarly rapid convergence For example since 2 1 253 1 024 the natural logarithm of 2 can be computed as ln 2 3 ln 1 1 4 ln 1 3 125 6 9 1 2 27 2 2 45 3 2 63 6 253 3 2 759 6 2 1265 9 2 1771 displaystyle begin aligned ln 2 amp 3 ln left 1 frac 1 4 right ln left 1 frac 3 125 right 8pt amp cfrac 6 9 cfrac 1 2 27 cfrac 2 2 45 cfrac 3 2 63 ddots cfrac 6 253 cfrac 3 2 759 cfrac 6 2 1265 cfrac 9 2 1771 ddots end aligned Furthermore since 10 1 2510 1 0243 even the natural logarithm of 10 can be computed similarly as ln 10 10 ln 1 1 4 3 ln 1 3 125 20 9 1 2 27 2 2 45 3 2 63 18 253 3 2 759 6 2 1265 9 2 1771 displaystyle begin aligned ln 10 amp 10 ln left 1 frac 1 4 right 3 ln left 1 frac 3 125 right 10pt amp cfrac 20 9 cfrac 1 2 27 cfrac 2 2 45 cfrac 3 2 63 ddots cfrac 18 253 cfrac 3 2 759 cfrac 6 2 1265 cfrac 9 2 1771 ddots end aligned The reciprocal of the natural logarithm can be also written in this way 1 ln x 2 x x 2 1 1 2 x 2 1 4 x 1 2 1 2 1 2 x 2 1 4 x displaystyle frac 1 ln x frac 2x x 2 1 sqrt frac 1 2 frac x 2 1 4x sqrt frac 1 2 frac 1 2 sqrt frac 1 2 frac x 2 1 4x ldots For example 1 ln 2 4 3 1 2 5 8 1 2 1 2 1 2 5 8 displaystyle frac 1 ln 2 frac 4 3 sqrt frac 1 2 frac 5 8 sqrt frac 1 2 frac 1 2 sqrt frac 1 2 frac 5 8 ldots Complex logarithms EditMain article Complex logarithm The exponential function can be extended to a function which gives a complex number as ez for any arbitrary complex number z simply use the infinite series with x z complex This exponential function can be inverted to form a complex logarithm that exhibits most of the properties of the ordinary logarithm There are two difficulties involved no x has ex 0 and it turns out that e2ip 1 e0 Since the multiplicative property still works for the complex exponential function ez ez 2kip for all complex z and integers k So the logarithm cannot be defined for the whole complex plane and even then it is multi valued any complex logarithm can be changed into an equivalent logarithm by adding any integer multiple of 2ip at will The complex logarithm can only be single valued on the cut plane For example ln i ip 2 or 5ip 2 or 3ip 2 etc and although i4 1 4 ln i can be defined as 2ip or 10ip or 6ip and so on Plots of the natural logarithm function on the complex plane principal branch z Re ln x yi z Im ln x yi z ln x yi Superposition of the previous three graphsSee also EditApproximating natural exponents log base e Iterated logarithm Napierian logarithm List of logarithmic identities Logarithm of a matrix Logarithmic differentiation Logarithmic integral function Nicholas Mercator first to use the term natural logarithm Polylogarithm Von Mangoldt functionNotes Edit Including C C SAS MATLAB Mathematica Fortran and some BASIC dialects For a similar approach to reduce round off errors of calculations for certain input values see trigonometric functions like versine vercosine coversine covercosine haversine havercosine hacoversine hacovercosine exsecant and excosecant References Edit G H Hardy and E M Wright An Introduction to the Theory of Numbers 4th Ed Oxford 1975 footnote to paragraph 1 7 log x is of course the Naperian logarithm of x to base e Common logarithms have no mathematical interest Mortimer Robert G 2005 Mathematics for physical chemistry 3rd ed Academic Press p 9 ISBN 0 12 508347 5 Extract of page 9 a b c Weisstein Eric W Natural Logarithm mathworld wolfram com Retrieved 2020 08 29 a b Rules Examples amp Formulas Logarithm Encyclopedia Britannica Retrieved 2020 08 29 Burn R P 2001 Alphonse Antonio de Sarasa and Logarithms Historia Mathematica pp 28 1 17 O Connor J J Robertson E F September 2001 The number e The MacTutor History of Mathematics archive Retrieved 2009 02 02 a b Cajori Florian 1991 A History of Mathematics 5th ed AMS Bookstore p 152 ISBN 0 8218 2102 4 An improper integral representation of the natural logarithm retrieved 2022 09 24 Logarithmic Expansions at Math2 org Leonhard Euler Introductio in Analysin Infinitorum Tomus Primus Bousquet Lausanne 1748 Exemplum 1 p 228 quoque in Opera Omnia Series Prima Opera Mathematica Volumen Octavum Teubner 1922 RUFFA Anthony A PROCEDURE FOR GENERATING INFINITE SERIES IDENTITIES PDF International Journal of Mathematics and Mathematical Sciences International Journal of Mathematics and Mathematical Sciences Retrieved 2022 02 27 Page 3654 equation 2 6 For a detailed proof see for instance George B Thomas Jr and Ross L Finney Calculus and Analytic Geometry 5th edition Addison Wesley 1979 Section 6 5 pages 305 306 OEIS A002392 Sasaki T Kanada Y 1982 Practically fast multiple precision evaluation of log x Journal of Information Processing 5 4 247 250 Retrieved 2011 03 30 Ahrendt Timm 1999 Fast Computations of the Exponential Function Stacs 99 Lecture Notes in Computer Science 1564 302 312 doi 10 1007 3 540 49116 3 28 ISBN 978 3 540 65691 3 Borwein Jonathan M Borwein Peter B 1987 Pi and the AGM A Study in Analytic Number Theory and Computational Complexity First ed Wiley Interscience ISBN 0 471 83138 7 page 225 Beebe Nelson H F 2017 08 22 Chapter 10 4 Logarithm near one The Mathematical Function Computation Handbook Programming Using the MathCW Portable Software Library 1 ed Salt Lake City UT USA Springer International Publishing AG pp 290 292 doi 10 1007 978 3 319 64110 2 ISBN 978 3 319 64109 6 LCCN 2017947446 S2CID 30244721 In 1987 Berkeley UNIX 4 3BSD introduced the log1p function a b c d Beebe Nelson H F 2002 07 09 Computation of expm1 exp x 1 PDF 1 00 Salt Lake City Utah USA Department of Mathematics Center for Scientific Computing University of Utah Retrieved 2015 11 02 a b c d HP 48G Series Advanced User s Reference Manual AUR 4 ed Hewlett Packard December 1994 1993 HP 00048 90136 0 88698 01574 2 Retrieved 2015 09 06 a b c d HP 50g 49g 48gII graphing calculator advanced user s reference manual AUR 2 ed Hewlett Packard 2009 07 14 2005 HP F2228 90010 Retrieved 2015 10 10 Searchable PDF Retrieved from https en wikipedia org w index php title Natural logarithm amp oldid 1132861462, wikipedia, wiki, book, books, library,

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