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logarithm
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{{short description|Inverse function of exponentiation that also maps products to sums}}{{Calculation results}}(File:Logarithm plots.png|right|thumb|upright=1.35|Plots of logarithm functions of three commonly used bases. The special points {{math|logb b {{=}} 1}} are indicated by dotted lines, and all curves intersect {{nowrap|in {{math|1= logb 1 = 0.}}}})File:Binary logarithm plot with ticks.svg|right|thumb|upright=1.35|alt=Graph showing a logarithmic curve, crossing the x-axis at x= 1 and approaching minus infinity along the y-axis.|The graph of the logarithm to base 2 crosses the x-axis at {{math|x {{=}} 1}} and passes through the points {{nowrap|(2, 1)}}, {{nowrap|(4, 2)}}, and {{nowrap|(8, 3)}}, depicting, e.g., {{math|log2(8) {{=}} 3}} and {{math|23 {{=}} 8}}. The graph gets arbitrarily close to the {{mvar|y}}-axis, but does not meet it.]]In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a given number {{mvar|x}} is the exponent to which another fixed number, the base {{mvar|b}}, must be raised, to produce that number {{mvar|x}}. In the simplest case, the logarithm counts the number of occurrences of the same factor in repeated multiplication; e.g., since {{math|1000 {{=}} 10â€‰Ã—â€‰10â€‰Ã—â€‰10 {{=}} 103}}, the "logarithm to base {{math|10}}" of {{math|1000}} is {{math|3}}. The logarithm of {{mvar|x}} to base {{mvar|b}} is denoted as {{math|logb (x)}}, or without parentheses, {{math|logb x}}, or even without the explicit base, {{math|log x}}â€”if no confusion is possible. More generally, exponentiation allows any positive real number as base to be raised to any real power, always producing a positive result, so {{math|logb (x)}} for any two positive real numbers {{mvar|b}} and {{mvar|x}}, where {{mvar|b}} is not equal to {{math|1}}, is always a unique real number {{mvar|y}}. More explicitly, the defining relation between exponentiation and logarithm is:
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log_b(x) = y quad exactly if quad b^y = x.
For example, {{math|1=log2 64 = 6}}, as {{math|1=26 = 64}}.The logarithm to base {{math|10}} (that is {{math|1=b = 10}}) is called the common logarithm and has many applications in science and engineering. The natural logarithm has the number {{mvar|e}} (that is {{math|b â‰ˆ 2.718}}) as its base; its use is widespread in mathematics and physics, because of its simpler integral and derivative. The binary logarithm uses base {{math|2}} (that is {{math|1=b = 2}}) and is commonly used in computer science. Logarithms are examples of concave functions.WEB,weblink The Ultimate Guide to Logarithm â€” Theory & Applications, 2016-05-08, Math Vault, en-US, 2019-07-24, Logarithms were introduced by John Napier in 1614 as a means to simplify calculations.BOOK,weblink John Napier and the invention of logarithms, 1614; a lecture, Hobson, Ernest William, 1914, Cambridge : University Press, University of California Libraries, 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 is possible because of the factâ€”important in its own rightâ€”that the logarithm of a product is the sum of the logarithms of the factors:
log_b(xy) = log_b x + log_b y, ,
provided that {{mvar|b}}, {{mvar|x}} and {{mvar|y}} are all positive and {{math|b â‰ 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 the exponential function in the 18th century, and who also introduced the letter {{mvar|e}} as the base of natural logarithms.BOOK,weblink Theory of complex functions, Remmert, Reinhold., 1991, Springer-Verlag, 0387971955, New York, 21118309, Logarithmic scales reduce wide-ranging quantities to tiny 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 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 describing 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.In the same way as the logarithm reverses exponentiation, the complex logarithm is the inverse function of the exponential function applied to complex numbers. The modular discrete logarithm is another variant; it has uses in public-key cryptography.Motivation and definition
Addition, multiplication, and exponentiation are three fundamental arithmetic operations (or in some context the first three hyperoperations). Addition, the simplest of these, can be undone by subtraction: for example, the addition of {{math|2}} in {{math|3 + 2 {{=}} 5}} can be undone by subtracting {{math|2: 5 âˆ’ 2 {{=}} 3}}. Multiplication, the next-simplest operation, can be undone by division: doubling a number {{mvar|x}}, i.e., multiplying {{mvar|x}} by {{math|2}}, the result is {{math|2x}}. To get back {{mvar|x}}, it is necessary to divide by {{math|2}}. For example, {{math|3 Ã— 2 {{=}} 6}} and the process of multiplying by {{math|2}} is undone by dividing by {{math|2}}: {{math|6 Ã· 2 {{=}} 3}}. The idea and purpose of logarithms is also to undo a fundamental arithmetic operation, namely raising a number to a certain power (an operation also known as exponentiation). For example, raising {{math|2}} to the power {{math|3}} yields {{math|8}}, because {{math|8}} is the product of three factors of {{math|2}}:
2^3 = 2 times 2 times 2 = 8
The logarithm (with respect to base {{math|2}}) of {{math|8}} is {{math|3}}, reflecting the fact that {{math|2}} was raised to the power {{math|3}} to get {{math|8}}.Exponentiation
This subsection contains a short overview of the exponentiation operation, which is fundamental to understanding logarithms.Raising {{mvar|b}} to the {{nowrap|{{mvar|n}}-th}} power, where {{mvar|n}} is a natural number, is done by multiplying {{mvar|n}} factors equal to {{mvar|b}}. The {{nowrap|{{mvar|n}}-th}} power of {{mvar|b}} is written {{math|bn}}, so that
b^n = underbrace{b times b times cdots times b}_{n text{ factors}}
Exponentiation may be extended to {{math|by}}, where {{mvar|b}} is a positive number and the exponent {{mvar|y}} is any real number.{{Citation|last1=Shirali| first1=Shailesh|title=A Primer on Logarithms|publisher=Universities Press|isbn=978-81-7371-414-6|year=2002|location=Hyderabad|url={{google books |plainurl=y |id=0b0igbb3WaQC}}}}, esp. section 2 For example, {{math|bâˆ’1}} is the reciprocal of {{mvar|b}}, that is, {{math|1/b}}. Raising {{mvar|b}} to the power 1/2 is the square root of {{mvar|b}}.More generally, raising {{mvar|b}} to a rational power {{math|p/q}}, where {{Mvar|p}} and {{Mvar|q}} are integers, is given by
b^{p / q} = sqrt[q]{b^p},
the {{Mvar|q}}-th root of b^p!!. Finally, any irrational number (a real number which is not rational) {{mvar|y}} can be approximated to arbitrary precision by rational numbers. This can be used to compute the {{mvar|y}}-th power of {{mvar|b}}: for example sqrt 2 approx 1.414 ... and b^{sqrt 2} is increasingly well approximated by b^1, b^{1.4}, b^{1.41}, b^{1.414}, .... A more detailed explanation, as well as the formula {{math|bm + n {{=}} {{mvar|b}}m Â· {{mvar|b}}{{mvar|n}}}} is contained in the article on exponentiation.Definition
The logarithm of a positive real number {{mvar|x}} with respect to base {{mvar|b}}{{refn|The restrictions on {{mvar|x}} and {{mvar|b}} are explained in the section "Analytic properties".|group=nb}} is the exponent by which {{mvar|b}} must be raised to yield {{mvar|x}}. In other words, the logarithm of {{mvar|x}} to base {{mvar|b}} is the solution {{mvar|y}} to the equation{{Citation|last1=Kate|first1=S.K.|last2=Bhapkar|first2=H.R.|title=Basics Of Mathematics|location=Pune|publisher=Technical Publications|isbn=978-81-8431-755-8|year=2009|url={{google books |plainurl=y |id=v4R0GSJtEQ4C|page=1}} }}, chapter 1
b^y = x.
The logarithm is denoted "{{math|logb x}}" (pronounced as "the logarithm of {{mvar|x}} to base {{mvar|b}}" or "the {{nowrap|base-b}} logarithm of {{mvar|x}}" or (most commonly) "the log, base {{mvar|b}}, of {{mvar|x}}").In the equation {{math|1=y = logb x}}, the value {{mvar|y}} is the answer to the question "To what power must {{mvar|b}} be raised, in order to yield {{mvar|x}}?".Examples
- {{math|log2 16 {{=}} 4 }}, since {{math| 24 {{=}} 2â€‰Ã—2â€‰Ã—â€‰2â€‰Ã—â€‰2 {{=}} 16}}.
- Logarithms can also be negative: quad log_2 ! frac{1}{2} = -1 quad since quad 2^{-1} = frac{1}{2^1} = frac{1}{2}.
- {{math|log10150}} is approximately 2.176, which lies between 2 and 3, just as 150 lies between {{math|102 {{=}} 100}} and {{math|103 {{=}} 1000.}}
- For any base {{mvar|b}}, {{math|logb {{mvar|b}} {{=}} 1}} and {{math|1= logb 1 = 0}}, since {{math|b1 {{=}} {{mvar|b}}}} and {{math|b0 {{=}} 1}}, respectively.
Logarithmic identities
Several important formulas, sometimes called logarithmic identities or logarithmic laws, relate logarithms to one another.All statements in this section can be found in {{Harvard citations|last1=Shirali|first1=Shailesh|year=2002|loc=section 4|nb=yes}}, {{Harvard citations|last1=Downing| first1=Douglas |year=2003|loc=p. 275}}, or {{Harvard citations|last1=Kate|last2=Bhapkar|year=2009|loc=p. 1-1|nb=yes}}, for example.Product, quotient, power, and root
The logarithm of 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 {{Mvar|p}}-th power of a number is {{Mvar|p}} times the logarithm of the number itself; the logarithm of a {{Mvar|p}}-th root is the logarithm of the number divided by {{Mvar|p}}. The following table lists these identities with examples. Each of the identities can be derived after substitution of the logarithm definitions x = b^{log_b x} or y = b^{log_b y} in the left hand sides.{| class="wikitable" style="margin: 0 auto;"! !! Formula !! Example| log_b(x y) = log_b x + log_b y| log_3 243 = log_3 (9 cdot 27) = log_3 9 + log_3 27 = 2 + 3 = 5 |
| log_b !frac{x}{y} = log_b x - log_b y| log_2 16 = log_2 !frac{64}{4} = log_2 64 - log_2 4 = 6 - 2 = 4 |
| log_bleft(x^pright) = p log_b x| log_2 64 = log_2 left(2^6right) = 6 log_2 2 = 6 |
| log_b sqrt[p]{x} = frac{log_b x}{p}| log_{10} sqrt{1000} = frac{1}{2}log_{10} 1000 = frac{3}{2} = 1.5 |
Change of base
The logarithm {{math|logbx}} can be computed from the logarithms of {{mvar|x}} and {{mvar|b}} with respect to an arbitrary base {{Mvar|k}} using the following formula:
log_b x = frac{log_k x}{log_k b}.,
{{Collapse top|title=Derivation of the conversion factor between logarithms of arbitrary base|width=80%}}Starting from the defining identity
x = b^{log_b x}
we can apply {{math|logk}} to both sides of this equation, to get
log_k x = log_k left(b^{log_b x}right) = log_b x cdot log_k b.
Solving for log_b x yields:
log_b x = frac{log_k x}{log_k b},
showing the conversion factor from given log_k-values to their corresponding log_b -values to be (log_k b)^{-1}.{{Collapse bottom}}Typical scientific calculators calculate the logarithms to bases 10 and {{mvar|e}}.{{Citation | last1=Bernstein | first1=Stephen | last2=Bernstein | first2=Ruth | title=Schaum's outline of theory and problems of elements of statistics. I, Descriptive statistics and probability | publisher=McGraw-Hill | location=New York | series=Schaum's outline series | isbn=978-0-07-005023-5 | year=1999 | url=https://archive.org/details/schaumsoutlineof00bern }}, p. 21 Logarithms with respect to any base {{mvar|b}} can be determined using either of these two logarithms by the previous formula:
log_b x = frac{log_{10} x}{log_{10} b} = frac{log_{e} x}{log_{e} b}. ,
Given a number {{mvar|x}} and its logarithm {{math|logbx}} to an unknown base {{mvar|b}}, the base is given by:
b = x^frac{1}{log_b x},
which can be seen from taking the defining equation x = b^{log_b x} to the power of ; tfrac{1}{log_b x}.Particular bases
(File:Log4.svg|thumb|upright=1.2|Plots of logarithm for bases 0.5, 2, and {{mvar|e}})Among all choices for the base, three are particularly common. These are {{math|1=b = 10}}, {{math|1=b = e}} (the irrational mathematical constant â‰ˆ 2.71828), and {{math|1=b = 2}} (the binary logarithm). In mathematical analysis, the logarithm to base {{mvar|e}} is widespread because of its particular analytical properties explained below. On the other hand, {{nowrap|base-10}} logarithms are easy to use for manual calculations in the decimal number system:{{Citation|last1=Downing|first1=Douglas|title=Algebra the Easy Way|series=Barron's Educational Series|location=Hauppauge, NY|publisher=Barron's|isbn=978-0-7641-1972-9|year=2003|url=https://archive.org/details/algebraeasyway00down_0}}, chapter 17, p. 275
log_{10}(10 x) = log_{10} 10 + log_{10} x = 1 + log_{10} x.
Thus, {{math|log10x}} is related to the number of decimal digits of a positive integer {{mvar|x}}: the number of digits is the smallest integer strictly bigger than log10x.{{Citation|last1=Wegener|first1=Ingo| title=Complexity theory: exploring the limits of efficient algorithms|publisher=Springer-Verlag|location=Berlin, New York|isbn=978-3-540-21045-0|year=2005}}, p. 20 For example, {{math|log101430}} is approximately 3.15. The next integer is 4, which is the number of digits of 1430. Both the natural logarithm and the logarithm to base two are used in information theory, corresponding to the use of nats or bits as the fundamental units of information, respectively.{{citation|title=Information Theory|first=Jan C. A.|last=Van der Lubbe|publisher=Cambridge University Press|year=1997|isbn=978-0-521-46760-5|page=3|url={{google books |plainurl=y |id=tBuI_6MQTcwC|page=3}}}} 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 cent is the binary logarithm (scaled by 1200) of the ratio between two adjacent equally-tempered pitches in European classical music, and in photography to measure exposure values.{{citation|title=The Manual of Photography|first1=Elizabeth|last1=Allen|first2=Sophie|last2=Triantaphillidou|publisher=Taylor & Francis|year=2011|isbn=978-0-240-52037-7|page=228|url={{google books |plainurl=y |id=IfWivY3mIgAC|page=228}}}}The following table lists common notations for logarithms to these bases and the fields where they are used. Many disciplines write {{math|logx}} instead of {{math|logbx}}, when the intended base can be determined from the context. The notation {{math|blogx}} also occurs.{{Citation| url=http://www.mathe-online.at/mathint/lexikon/l.html |author1=Franz Embacher |author2=Petra Oberhuemer |title=Mathematisches Lexikon |publisher=mathe online: fÃ¼r Schule, Fachhochschule, UniversitÃ¤t unde Selbststudium |accessdate=2011-03-22 |language=German}} The "ISO notation" column lists designations suggested by the International Organization for Standardization (ISO 31-11).{{Citation|title = Guide for the Use of the International System of Units (SI)|first = B.N.|last = Taylor|publisher = US Department of Commerce|year = 1995|url =weblink|access-date = 25 May 2007|archive-url =weblink" title="web.archive.org/web/20070629210131weblink">weblink|archive-date = 29 June 2007|dead-url = yes|df = dmy-all}} Because the notation {{math|log {{mvar|x}}}} has been used for all three bases (or when the base is indeterminate or immaterial), the intended base must often be inferred based on context or discipline. In computer science and mathematics, {{Math|log}} usually refers to {{math|log2}} and {{math|loge}}, respectively.{{citation|first1=Michael T.|last1=Goodrich|author1-link=Michael T. Goodrich|first2=Roberto|last2=Tamassia|author2-link=Roberto Tamassia|title=Algorithm Design: Foundations, Analysis, and Internet Examples|publisher=John Wiley & Sons|year=2002|page=23|quote=One of the interesting and sometimes even surprising aspects of the analysis of data structures and algorithms is the ubiquitous presence of logarithms ... As is the custom in the computing literature, we omit writing the base {{mvar|b}} of the logarithm when {{math|1=b = 2}}.}} In other contexts {{Math|log}} often means {{math|log10}}.BOOK, Introduction to Applied Mathematics for Environmental Science, illustrated, David F., Parkhurst, Springer Science & Business Media, 2007, 978-0-387-34228-3, 288, {{google books, y, h6yq_lOr8Z4C, 288, }}{| class="wikitable" style="text-align:center; margin:1em auto 1em auto;"! scope="col"|Base {{mvar|b}}! scope="col"|Name for logbx! scope="col"|ISO notation! scope="col"|Other notations! scope="col"|Used inlb {{mvar | title = Mathematics: from the birth of numbers. | location=New York | year = 1997|isbn=978-0-393-04002-9}} | ld {{mvar | log {{mvar | lg {{mvar | LAST2=REINGOLD | TITLE=UNDERSTANDING THE COMPLEXITY OF INTERPOLATION SEARCH | DATE=DECEMBER 1977 | ISSUE=6 | DOI=10.1016/0020-0190(77)90072-2, {{math|log2x}}| computer science, information theory, music theory, photography |
ln {{mvar | Some mathematicians disapprove of this notation. In his 1985 autobiography, Paul Halmos criticized what he considered the "childish ln notation," which he said no mathematician had ever used.{{Citation|title = I Want to Be a Mathematician: An Automathography|author = Paul Halmos|publisher = Springer-Verlag|location=Berlin, New York|year = 1985|isbn=978-0-387-96078-4}}The notation was invented by Irving Stringham, a mathematician.{{Citation|title = Uniplanar algebra: being part I of a propÃ¦deutic to the higher mathematical analysis|author = Irving Stringham|publisher = The Berkeley Press|year = 1893|page = xiii | plainurl=y | page=13}}}}{{Citation|title = Introduction to Financial Technology|author = Roy S. Freedman|publisher = Academic Press|location=Amsterdam|year = 2006|isbn=978-0-12-370478-8|page = 59|url = {{google books |plainurl=y |id=APJ7QeR_XPkC|page=5}}}}|name=adaa|group=nb}} | log {{mvar | TITLE=PRINCIPLES OF MATHEMATICAL ANALYSIS | PUBLISHER=MCGRAW-HILL INTERNATIONAL | ISBN=978-0-07-085613-4 | URL=HTTPS://ARCHIVE.ORG/DETAILS/PRINCIPLESOFMATH00RUDI, and many programming languages{{refn | C (programming language)>C, Java (programming language) | , Haskell (programming language)>Haskell, and BASIC programming language | .>group=nb}})| mathematics, physics, chemistry,statistics, economics, information theory, and engineering |
lg {{mvar|x}}}} | log {{mvar | log10x}}(in engineering, biology, astronomy) | engineering fields (see decibel and see below), logarithm Mathematical table>tables, handheld calculators, spectroscopy |
History
The history of logarithm in seventeenth-century Europe is 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 Rule of Logarithms).{{citation |first=John |last=Napier |author-link=John Napier |title=Mirifici Logarithmorum Canonis Descriptio |trans-title=The Description of the Wonderful Rule of Logarithms |language=Latin |location=Edinburgh, Scotland |publisher=Andrew Hart |year=1614 |url=http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN527914568&DMDID=DMDLOG_0001&LOGID=LOG_0001&PHYSID=PHYS_0001 }}{{Citation|first=Ernest William |last=Hobson|title=John Napier and the invention of logarithms, 1614|year=1914|publisher=The University Press|location=Cambridge|url=https://archive.org/details/johnnapierinvent00hobsiala}} Prior to Napier's invention, there had been other techniques of similar scopes, such as the prosthaphaeresis or the use of tables of progressions, extensively developed by Jost BÃ¼rgi around 1600.{{citation |first1=Menso |last1=Folkerts |first2=Dieter |last2=Launert |first3=Andreas |last3=Thom |date=October 2015 |title=Jost BÃ¼rgi's Method for Calculating Sines |arxiv=1510.03180|bibcode=2015arXiv151003180F }}WEB,weblink Burgi biography, www-history.mcs.st-and.ac.uk, 2018-02-14, The common logarithm of a number is the index of that power of ten which equals the number.William Gardner (1742) Tables of Logarithms 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".R.C. Pierce (1977) "A brief history of logarithm", Two-Year College Mathematics Journal 8(1):22â€“26. 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.Enrique Gonzales-Velasco (2011) Journey through Mathematics â€“ Creative Episodes in its History, Â§2.4 Hyperbolic logarithms, p. 117, Springer {{isbn|978-0-387-92153-2}}Such methods are called prosthaphaeresis.Invention of the function now known as natural logarithm began as an attempt to perform a quadrature of a rectangular hyperbola by GrÃ©goire 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 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, Patavii, and James Gregory. The notation Log y was adopted by Leibniz in 1675,Florian Cajori (1913) "History of the exponential and logarithm concepts", American Mathematical Monthly 20: 5, 35, 75, 107, 148, 173, 205. and the next year he connected it to the integralint frac{dy}{y} .Logarithm tables, slide rules, and historical applications{{anchor|Antilogarithm}}
missing image!
- Logarithms Britannica 1797.png -
The 1797 EncyclopÃ¦dia Britannica explanation of logarithms
By simplifying difficult calculations before calculators and computers became available, logarithms contributed to the advance of science, especially astronomy. They were critical to advances in surveying, celestial navigation, and other domains. Pierre-Simon Laplace called logarithms
- Logarithms Britannica 1797.png -
The 1797 EncyclopÃ¦dia Britannica explanation of logarithms
"...[a]n admirable artifice which, by reducing to a few days the labour of many months, doubles the life of the astronomer, and spares him the errors and disgust inseparable from long calculations."{{Citation |last1=Bryant |first1=Walter W. |title=A History of Astronomy |url=https://archive.org/stream/ahistoryastrono01bryagoog#page/n72/mode/2up |publisher=Methuen & Co|location=London |year=1907 }}, p. 44
Log tables
A key tool that enabled the practical use of logarithms was the table of logarithms.{{Citation | last1=Campbell-Kelly | first1=Martin | title=The history of mathematical tables: from Sumer to spreadsheets | publisher=Oxford University Press | series=Oxford scholarship online | isbn=978-0-19-850841-0 | year=2003}}, section 2 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 1â€“1000, with a precision of 14 digits. Subsequently, tables with increasing scope were written. These tables listed the values of {{math|log10x}} for any number {{mvar|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 {{mvar|x}} can be separated into an integer part and a fractional part, known as the characteristic and mantissa. Tables of logarithms need only include the mantissa, as the characteristic can be easily determined by counting digits from the decimal point.{{Citation | last1=Spiegel | first1=Murray R. | last2=Moyer | first2=R.E. | title=Schaum's outline of college algebra | publisher=McGraw-Hill | location=New York | series=Schaum's outline series | isbn=978-0-07-145227-4 | year=2006}}, p. 264 The characteristic of {{math|10 Â· {{mvar|x}}}} is one plus the characteristic of {{mvar|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 cdot 3.542) = 3 + log_{10}3.542 approx 3 + log_{10}3.54 ,
Greater accuracy can be obtained by interpolation:
log_{10}3542 approx 3 + log_{10}3.54 + 0.2 (log_{10}3.55-log_{10}3.54),
The value of {{math| 10x}} can be determined by reverse look up in the same table, since the logarithm is a monotonic function.Computations
The product and quotient of two positive numbers {{Mvar|c}} and {{Mvar|d}} were routinely calculated as the sum and difference of their logarithms. The product {{Mvar|cd}} or quotient {{Mvar|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} ,
and
frac c d = c d^{-1} = 10^{ log_{10} c - log_{10} d}. ,
For manual calculations that demand any appreciable precision, performing 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 look-ups by
c^d = left(10^{ log_{10} c}right)^d = 10^{d log_{10} c} ,
and
sqrt[d]{c} = c^frac{1}{d} = 10^{frac{1}{d} log_{10} c}. ,
Trigonometric calculations were facilitated by tables that contained the common logarithms of trigonometric functions.Slide rules
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 ruleâ€”a pair of logarithmic scales movable with respect to each other. Numbers are placed on sliding scales at distances proportional to the differences between their logarithms. Sliding the upper scale appropriately amounts to mechanically adding logarithms, as illustrated here:center|thumb|550px|Schematic depiction of a slide rule. Starting from 2 on the lower scale, add the distance to 3 on the upper scale to reach the product 6. The slide rule works because it is marked such that the distance from 1 to {{mvar|x}} is proportional to the logarithm of {{mvar|x}}.|alt=A slide rule: two rectangles with logarithmically ticked axes, arrangement to add the distance from 1 to 2 to the distance from 1 to 3, indicating the product 6.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 the 1970s, because it allows, at the expense of precision, much faster computation than techniques based on tables.{hide}Harvard citations
|last1=Maor |year=2009 |nb=yes |loc=sections 1, 13{edih}
Analytic properties
A deeper study of logarithms requires the concept of a function. A function is a rule that, given one number, produces another number.BOOK, Devlin, Keith, Keith Devlin, Sets, functions, and logic: an introduction to abstract mathematics, Chapman & Hall/CRC, Boca Raton, Fla, 3rd, Chapman & Hall/CRC mathematics, 978-1-58488-449-1, 2004, {{google books, y, uQHF7bcm4k4C, }}, or see the references in function An example is the function producing the {{mvar|x}}-th power of {{mvar|b}} from any real number {{mvar|x}}, where the base {{mvar|b}} is a fixed number. This function is written: f(x) = b^x. ,Logarithmic function
To justify the definition of logarithms, it is necessary to show that the equation
b^x = y ,
has a solution {{mvar|x}} and that this solution is unique, provided that {{mvar|y}} is positive and that {{mvar|b}} is positive and unequal to 1. A proof of that fact requires the intermediate value theorem from elementary calculus.{{Citation|last1=Lang|first1=Serge|author1-link=Serge Lang|title=Undergraduate analysis|publisher=Springer-Verlag|location=Berlin, New York|edition=2nd|series=Undergraduate Texts in Mathematics|isbn=978-0-387-94841-6|mr=1476913|year=1997|doi=10.1007/978-1-4757-2698-5}}, section III.3 This theorem states that a continuous function that produces two values {{mvar|m}} and {{mvar|n}} also produces any value that lies between {{mvar|m}} and {{mvar|n}}. A function is continuous if it does not "jump", that is, if its graph can be drawn without lifting the pen.This property can be shown to hold for the function {{math|1=f(x) = {{mvar|b}}x}}. Because {{mvar|f}} takes arbitrarily large and arbitrarily small positive values, any number {{math|y > 0}} lies between {{math|f(x0)}} and {{math|f(x1)}} for suitable {{math|x0}} and {{math|x1}}. Hence, the intermediate value theorem ensures that the equation {{math|1=f(x) = {{mvar|y}}}} has a solution. Moreover, there is only one solution to this equation, because the function f is strictly increasing (for {{math|b > 1}}), or strictly decreasing (for {{math|0 < {{mvar|b}} < 1}}).{{Harvard citations|last1=Lang|year=1997 |nb=yes|loc=section IV.2}}The unique solution {{mvar|x}} is the logarithm of {{mvar|y}} to base {{mvar|b}}, {{math|logby}}. The function that assigns to {{mvar|y}} its logarithm is called logarithm function or logarithmic function (or just logarithm).The function {{math|logbx}} is essentially characterized by the above product formula
log_b(xy) = log_b x + log_b y.
More precisely, the logarithm to any base {{math|b > 1}} is the only increasing function f from the positive reals to the reals satisfying {{math|1=f(b) = 1}} and BOOK, Foundations of Modern Analysis, 1, DieudonnÃ©, Jean, 84, 1969, Academic Press, item (4.3.1)
f(xy)=f(x)+f(y).
Inverse function
File:Logarithm inversefunctiontoexp.svg|right|thumb|The graph of the logarithm function {{math|logb(x)}} (blue) is obtained by reflectingreflectingThe formula for the logarithm of a power says in particular that for any number {{mvar|x}},
log_b left (b^x right) = x log_b b = x.
In prose, taking the {{math|x-th}} power of {{mvar|b}} and then the {{math|base-b}} logarithm gives back {{mvar|x}}. Conversely, given a positive number {{mvar|y}}, the formula
b^{log_b y} = y
says that first taking the logarithm and then exponentiating gives back {{mvar|y}}. Thus, the two possible ways of combining (or composing) logarithms and exponentiation give back the original number. Therefore, the logarithm to base {{mvar|b}} is the inverse function of {{math|f(x) {{=}} {{mvar|b}}x}}.{{Citation | last1=Stewart | first1=James | title=Single Variable Calculus: Early Transcendentals | publisher=Thomson Brooks/Cole |location=Belmont|isbn=978-0-495-01169-9 | year=2007}}, section 1.6Inverse functions are closely related to the original functions. Their graphs correspond to each other upon exchanging the {{mvar|x}}- and the {{mvar|y}}-coordinates (or upon reflection at the diagonal line {{mvar|x}} = {{mvar|y}}), as shown at the right: a point {{math|1=(t, u = {{mvar|b}}t)}} on the graph of f yields a point {{math|1=(u, t = logb'u)}} on the graph of the logarithm and vice versa. As a consequence, {{math|logb(x)}} diverges to infinity (gets bigger than any given number) if {{mvar|x}} grows to infinity, provided that {{mvar|b}} is greater than one. In that case, {{math|logb(x)}} is an increasing function. For {{math|b < 1}}, {{math|logb(x)}} tends to minus infinity instead. When {{mvar|x}} approaches zero, {{math|logb'x}} goes to minus infinity for {{math|b > 1}} (plus infinity for {{math|b < 1}}, respectively).Derivative and antiderivative
File:Logarithm derivative.svg|right|thumb|220px|The graph of the x {{=}} 1.5}} (black)|alt=A graph of the logarithm function and a line touching it in one point.Analytic properties of functions pass to their inverses. Thus, as {{math|1=f(x) = {{mvar|b}}x}} is a continuous and differentiable function, so is {{math|logb'y}}. Roughly, a continuous function is differentiable if its graph has no sharp "corners". Moreover, as the derivative of {{math|f(x)}} evaluates to {{math|ln(b)b'x}} by the properties of the exponential function, the chain rule implies that the derivative of {{math|logbx}} is given byWEB, Wolfram Alpha, Calculation of d/dx(Log(b,x)), Wolfram Research, 15 March 2011,weblink
frac{d}{dx} log_b x = frac{1}{xln b}.
That is, the slope of the tangent touching the graph of the {{math|base-b}} logarithm at the point {{math|(x, logb(x))}} equals {{math|1/(xâ€‰ln(b))}}.The derivative of ln {{mvar|x}} is 1/x; this implies that ln {{mvar|x}} is the unique antiderivative of {{math|1/x}} that has the value 0 for {{math|1=x =1}}. It is this very simple formula that motivated to qualify as "natural" the natural logarithm; this is also one of the main reasons of the importance of the constant {{mvar|e}}.The derivative with a generalised functional argument {{math|f(x)}} is
frac{d}{dx} ln f(x) = frac{f'(x)}{f(x)}.
The quotient at the right hand side is called the logarithmic derivative of f. Computing {{math|f'(x)}} by means of the derivative of {{math|ln(f(x))}} is known as logarithmic differentiation.{{Citation | last1=Kline | first1=Morris | author1-link=Morris Kline | title=Calculus: an intuitive and physical approach | publisher=Dover Publications | location=New York | series=Dover books on mathematics | isbn=978-0-486-40453-0 | year=1998}}, p. 386 The antiderivative of the natural logarithm {{math|ln(x)}} is:WEB, Wolfram Alpha, Calculation of Integrate(ln(x)), Wolfram Research, 15 March 2011,weblink
int ln(x) ,dx = x ln(x) - x + C.
Related formulas, such as antiderivatives of logarithms to other bases can be derived from this equation using the change of bases.{{Harvard citations|editor1-last=Abramowitz|editor2-last=Stegun|year=1972 |nb=yes|loc=p. 69}}Integral representation of the natural logarithm
File:Natural logarithm integral.svg|right|thumb|The t}} is the shaded area underneath the graph of the function {{math|1=f(x) = 1/x}} (reciprocal of {{mvar|x}}).|alt=A hyperbola with part of the area underneath shaded in grey.The natural logarithm of {{Mvar|t}} equals the integral of {{Mvar|1/x}} {{Mvar|dx}} from 1 to {{Mvar|t}}:
ln (t) = int_1^t frac{1}{x} , dx.
In other words, {{math|ln(t)}} equals the area between the {{mvar|x}} axis and the graph of the function {{math|1/x}}, ranging from {{math|1=x = 1}} to {{math|1=x = t}} (figure at the right). This is a consequence of the fundamental theorem of calculus and the fact that the derivative of {{math|ln(x)}} is {{math|1/x}}. The right hand side of this equation can serve as a definition of the natural logarithm. Product and power logarithm formulas can be derived from this definition.{{Citation|last1=Courant|first1=Richard|title=Differential and integral calculus. Vol. I|publisher=John Wiley & Sons|location=New York|series=Wiley Classics Library|isbn=978-0-471-60842-4|mr=1009558|year=1988}}, section III.6 For example, the product formula {{math|1=ln(tu) = ln(t) + ln(u)}} is deduced as:
ln(tu) = int_1^{tu} frac{1}{x} , dx stackrel {(1)} = int_1^ frac{1}{x} , dx + int_t^{tu} frac{1}{x} , dx stackrel {(2)} = ln(t) + int_1^u frac{1}{w} , dw = ln(t) + ln(u).
The equality (1) splits the integral into two parts, while the equality (2) is a change of variable ({{math|1=w = {{mvar|x}}/t}}). In the illustration below, the splitting corresponds to dividing the area into the yellow and blue parts. Rescaling the left hand blue area vertically by the factor t and shrinking it by the same factor horizontally does not change its size. Moving it appropriately, the area fits the graph of the function {{math|1=f(x) = 1/x}} again. Therefore, the left hand blue area, which is the integral of {{math|f(x)}} from t to tu is the same as the integral from 1 to u. This justifies the equality (2) with a more geometric proof.(File:Natural logarithm product formula proven geometrically.svg|thumb|center|500px|A visual proof of the product formula of the natural logarithm|alt=The hyperbola depicted twice. The area underneath is split into different parts.)The power formula {{math|1=ln(tr) = r ln(t)}} may be derived in a similar way:
1 + frac 1 2 + frac 1 3 + cdots + frac 1 n = sum_{k=1}^n frac{1}{k},
is called the harmonic series. It is closely tied to the natural logarithm: as n tends to infinity, the difference,
sum_{k=1}^n frac{1}{k} - ln(n),
converges (i.e., gets arbitrarily close) to a number known as the Eulerâ€“Mascheroni constant {{math|1 = Î³ = 0.5772...}}. This relation aids in analyzing the performance of algorithms such as quicksort.{{Citation|last1=Havil|first1=Julian|title=Gamma: Exploring Euler's Constant|publisher=Princeton University Press|isbn=978-0-691-09983-5|year=2003}}, sections 11.5 and 13.8There are also some other integral representations of the logarithm that are useful in some situations:
ln(x) = -lim_{epsilon to 0} int_epsilon^infty frac{dt}left( e^{-xt} - e^{-t} right)
ln(x) = int_0^infty,frac{dt},left[cos(t)-cos(xt)right]
The first identity can be verified by showing that it has the same value at {{math|1=x = 1}}, and the same derivative.The second identity can be proven by writing
frac1t =int_0^infty,dq,e^{-qt}
and then inserting the Laplace transform of {{math|1=cos(xt)}} (and {{math|1=cos(t)}}).Transcendence of the logarithm
Real numbers that are not algebraic are called transcendental;{{citation|title=Selected papers on number theory and algebraic geometry|volume=172|first1=Katsumi|last1=Nomizu|authorlink=Katsumi Nomizu|location=Providence, RI|publisher=AMS Bookstore|year=1996|isbn=978-0-8218-0445-2|page=21|url={{google books |plainurl=y |id=uDDxdu0lrWAC|page=21}}}} for example, {{pi}} and e are such numbers, but sqrt{2-sqrt 3} is not. Almost all real numbers are transcendental. The logarithm is an example of a transcendental function. The Gelfondâ€“Schneider theorem asserts that logarithms usually take transcendental, i.e., "difficult" values.{{Citation|last1=Baker|first1=Alan|author1-link=Alan Baker (mathematician)|title=Transcendental number theory|publisher=Cambridge University Press|isbn=978-0-521-20461-3|year=1975}}, p. 10Calculation
(File:Logarithm keys.jpg|thumb|The logarithm keys (LOG for base-10 and LN for base-{{mvar|e}}) on a typical scientific calculator)Logarithms are easy to compute in some cases, such as {{math|1=log10(1000) = 3}}. In general, logarithms can be calculated using power series or the arithmeticâ€“geometric mean, or be retrieved from a precalculated logarithm table that provides a fixed precision.{{Citation | last1=Muller | first1=Jean-Michel | title=Elementary functions | publisher=BirkhÃ¤user Boston | location=Boston, MA | edition=2nd | isbn=978-0-8176-4372-0 | year=2006}}, sections 4.2.2 (p. 72) and 5.5.2 (p. 95){{Citation |last1=Hart |last2=Cheney |last3=Lawson |year=1968|publisher=John Wiley|location=New York|title=Computer Approximations|series=SIAM Series in Applied Mathematics|display-authors=etal}}, section 6.3, pp. 105â€“11Newton's method, an iterative method to solve equations approximately, can also be used to calculate the logarithm, because its inverse function, the exponential function, can be computed efficiently.{{Citation|last1=Zhang |first1=M. |last2=Delgado-Frias |first2=J.G. |last3=Vassiliadis |first3=S. |title=Table driven Newton scheme for high precision logarithm generation |url=http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=326783 |doi=10.1049/ip-cdt:19941268 |journal=IEE Proceedings Computers & Digital Techniques |issn=1350-2387 |volume=141 |year=1994 |issue=5 |pages=281â€“92 |deadurl=unfit |archiveurl=https://web.archive.org/web/20150529214127weblink |archivedate=29 May 2015 }}, section 1 for an overview Using look-up tables, CORDIC-like methods can be used to compute logarithms if the only available operations are addition and bit shifts.{{Citation |url= |first=J.E.|last=Meggitt|title=Pseudo Division and Pseudo Multiplication Processes|journal=IBM Journal|date=April 1962|doi=10.1147/rd.62.0210|volume=6|issue=2|pages=210â€“26}}{{Citation |last=Kahan |first=W. |authorlink= William Kahan |title=Pseudo-Division Algorithms for Floating-Point Logarithms and Exponentials |date= May 20, 2001 |publisher= |journal= |doi= }} Moreover, the binary logarithm algorithm calculates {{math|lb(x)}} recursively based on repeated squarings of {{mvar|x}}, taking advantage of the relation
log_2left(x^2right) = 2 log_2 (x). ,
Power series
- Taylor series
- More efficient series
ln (z) = 2sum_{k=0}^inftyfrac{1}{2k+1}left(frac{z-1}{z+1}right)^{2k+1}.
This series can be derived from the above Taylor series. It converges more quickly than the Taylor series, especially if {{mvar|z}} is close to 1. For example, for {{math|1=z = 1.5}}, the first three terms of the second series approximate {{math|ln(1.5)}} with an error of about {{val|3|e=-6}}. The quick convergence for {{mvar|z}} close to 1 can be taken advantage of in the following way: given a low-accuracy approximation {{math|y â‰ˆ ln(z)}} and putting
A = frac z{exp(y)}, ,
the logarithm of {{mvar|z}} is:
ln (z)=y+ln (A). ,
The better the initial approximation {{mvar|y}} is, the closer {{mvar|A}} is to 1, so its logarithm can be calculated efficiently. {{mvar|A}} can be calculated using the exponential series, which converges quickly provided {{mvar|y}} is not too large. Calculating the logarithm of larger {{mvar|z}} can be reduced to smaller values of {{mvar|z}} by writing {{math|z {{=}} a Â· 10b}}, so that {{math|ln(z) {{=}} ln(a) + {{mvar|b}} Â· ln(10)}}.A closely related method can be used to compute the logarithm of integers. Putting textstyle z=frac{n+1}{n} in the above series, it follows that:
ln (n+1) = ln(n) + 2sum_{k=0}^inftyfrac{1}{2k+1}left(frac{1}{2 n+1}right)^{2k+1}.
If the logarithm of a large integer {{mvar|n}} is known, then this series yields a fast converging series for {{math|log(n+1)}}, with a rate of convergence of frac{1}{2 n+1}.Arithmeticâ€“geometric mean approximation
The arithmeticâ€“geometric mean yields high precision approximations of the natural logarithm. Sasaki and Kanada showed in 1982 that it was particularly fast for precisions between 400 and 1000 decimal places, while Taylor series methods were typically faster when less precision was needed. In their work {{math|ln(x)}} is approximated to a precision of {{math|2âˆ’p}} (or p precise bits) by the following formula (due to Carl Friedrich Gauss):{{Citation |first1=T. |last1=Sasaki |first2=Y. |last2=Kanada |title=Practically fast multiple-precision evaluation of log(x) |journal=Journal of Information Processing |volume=5|issue=4 |pages=247â€“50 |year=1982 | url=http://ci.nii.ac.jp/naid/110002673332 | accessdate=30 March 2011}}{{Citation |first1=Timm |title=Stacs 99|last1=Ahrendt|publisher=Springer|location=Berlin, New York|series=Lecture notes in computer science|doi=10.1007/3-540-49116-3_28|volume=1564|year=1999|pages=302â€“12|isbn=978-3-540-65691-3|chapter=Fast Computations of the Exponential Function}}
ln (x) approx frac{pi}{2 M(1,2^{2-m}/x)} - m ln (2).
Here {{math|M(x,y)}} denotes the arithmeticâ€“geometric mean of {{mvar|x}} and {{mvar|y}}. It is obtained by repeatedly calculating the average (x+y)/2 (arithmetic mean) and sqrt{xy} (geometric mean) of {{mvar|x}} and {{mvar|y}} then let those two numbers become the next {{mvar|x}} and {{mvar|y}}. The two numbers quickly converge to a common limit which is the value of {{math|M(x,y)}}. m is chosen such that
x ,2^m > 2^{p/2}.,
to ensure the required precision. A larger m makes the {{math|M(x,y)}} calculation take more steps (the initial x and y are farther apart so it takes more steps to converge) but gives more precision. The constants {{math|pi}} and {{math|ln(2)}} can be calculated with quickly converging series.Feynman's algorithm
While at Los Alamos National Laboratory working on the Manhattan Project, Richard Feynman developed a bit-processing algorithm that is similar to long division and was later used in the Connection Machine. The algorithm uses the fact that every real number 1 < x < 2 is uniquely representable as a product of distinct factors of the form 1 + 2^{-k} . The algorithm sequentially builds that product P: if P cdot (1 + 2^{-k}) < x, then it changes P to P cdot (1 + 2^{-k}) . It then increase k by one regardless. The algorithm stops when k is large enough to give the desired accuracy. Because log(x) is the sum of the terms of the form log(1 + 2^{-k}) corresponding to those k for which the factor 1 + 2^{-k} was included in the product P, log(x) may be computed by simple addition, using a table of log(1 + 2^{-k}) for all k. Any base may be used for the logarithm table.JOURNAL, Danny, Hillis, Danny Hillis, Richard Feynman and The Connection Machine, Physics Today, 42, 2, 78, January 15, 1989, 10.1063/1.881196, 1989PhT....42b..78H,Applications
File:NautilusCutawayLogarithmicSpiral.jpg|thumb|A alt=A photograph of a nautilus' shell.Logarithms have many applications inside and outside mathematics. Some of these occurrences are related to the notion of scale invariance. For example, each chamber of the shell of a nautilus is an approximate copy of the next one, scaled by a constant factor. This gives rise to a logarithmic spiral.{{Harvard citations
|last1=Maor
|year=2009
|nb=yes
|loc=p. 135
}} Benford's law on the distribution of leading digits can also be explained by scale invariance.{{Citation | last1=Frey | first1=Bruce | title=Statistics hacks | publisher=O'Reilly|location=Sebastopol, CA| series=Hacks Series |url={{google books |plainurl=y |id=HOPyiNb9UqwC|page=275}}| isbn=978-0-596-10164-0 | year=2006}}, chapter 6, section 64 Logarithms are also linked to self-similarity. For example, logarithms appear in the analysis of algorithms that solve a problem by dividing it into two similar smaller problems and patching their solutions.{{Citation | last1=Ricciardi | first1=Luigi M. | title=Lectures in applied mathematics and informatics | url={{google books |plainurl=y |id=Cw4NAQAAIAAJ}} | publisher=Manchester University Press | location=Manchester | isbn=978-0-7190-2671-3 | year=1990}}, p. 21, section 1.3.2 The dimensions of self-similar geometric shapes, that is, shapes whose parts resemble the overall picture are also based on logarithms.
Logarithmic scales are useful for quantifying the relative change of a value as opposed to its absolute difference. Moreover, because the logarithmic function {{math|log(x)}} grows very slowly for large {{mvar|x}}, logarithmic scales are used to compress large-scale scientific data. Logarithms also occur in numerous scientific formulas, such as the Tsiolkovsky rocket equation, the Fenske equation, or the Nernst equation.|year=2009
|nb=yes
|loc=p. 135
}} Benford's law on the distribution of leading digits can also be explained by scale invariance.{{Citation | last1=Frey | first1=Bruce | title=Statistics hacks | publisher=O'Reilly|location=Sebastopol, CA| series=Hacks Series |url={{google books |plainurl=y |id=HOPyiNb9UqwC|page=275}}| isbn=978-0-596-10164-0 | year=2006}}, chapter 6, section 64 Logarithms are also linked to self-similarity. For example, logarithms appear in the analysis of algorithms that solve a problem by dividing it into two similar smaller problems and patching their solutions.{{Citation | last1=Ricciardi | first1=Luigi M. | title=Lectures in applied mathematics and informatics | url={{google books |plainurl=y |id=Cw4NAQAAIAAJ}} | publisher=Manchester University Press | location=Manchester | isbn=978-0-7190-2671-3 | year=1990}}, p. 21, section 1.3.2 The dimensions of self-similar geometric shapes, that is, shapes whose parts resemble the overall picture are also based on logarithms.
Logarithmic scale
File:Germany Hyperinflation.svg|A logarithmic chart depicting the value of one Goldmark in Papiermarks during the German hyperinflation in the 1920sGerman hyperinflation in the 1920sScientific quantities are often expressed as logarithms of other quantities, using a logarithmic scale. For example, the decibel is a unit of measurement associated with logarithmic-scale quantities. It is based on the common logarithm of ratiosâ€”10 times the common logarithm of a power ratio or 20 times the common logarithm of a voltage ratio. It is used to quantify the loss of voltage levels in transmitting electrical signals,{{Citation|last1=Bakshi|first1=U.A.|title=Telecommunication Engineering |publisher=Technical Publications|location=Pune|isbn=978-81-8431-725-1|year=2009|url={{google books |plainurl=y |id=EV4AF0XJO9wC|page=A5}}}}, section 5.2 to describe power levels of sounds in acoustics,{{Citation|last1=Maling|first1=George C.|editor1-last=Rossing|editor1-first=Thomas D.|title=Springer handbook of acoustics|publisher=Springer-Verlag|location=Berlin, New York|isbn=978-0-387-30446-5|year=2007|chapter=Noise}}, section 23.0.2 and the absorbance of light in the fields of spectrometry and optics. The signal-to-noise ratio describing the amount of unwanted noise in relation to a (meaningful) signal is also measured in decibels.{{Citation | last1=Tashev | first1=Ivan Jelev | title=Sound Capture and Processing: Practical Approaches | publisher=John Wiley & Sons | location=New York | isbn=978-0-470-31983-3 | year=2009|url={{google books |plainurl=y |id=plll9smnbOIC|page=48}}|page=98}} In a similar vein, the peak signal-to-noise ratio is commonly used to assess the quality of sound and image compression methods using the logarithm.{{Citation | last1=Chui | first1=C.K. | title=Wavelets: a mathematical tool for signal processing | publisher=Society for Industrial and Applied Mathematics | location=Philadelphia | series=SIAM monographs on mathematical modeling and computation | isbn=978-0-89871-384-8 | year=1997|url={{google books |plainurl=y |id=N06Gu433PawC|page=180}}|page=}}The strength of an earthquake is measured by taking the common logarithm of the energy emitted at the quake. This is used in the moment magnitude scale or the Richter magnitude scale. For example, a 5.0 earthquake releases 32 times {{math|(101.5)}} and a 6.0 releases 1000 times {{math|(103)}} the energy of a 4.0.{{Citation|last1=Crauder|first1=Bruce|last2=Evans|first2=Benny|last3=Noell|first3=Alan|title=Functions and Change: A Modeling Approach to College Algebra|publisher=Cengage Learning|location=Boston|edition=4th|isbn=978-0-547-15669-9|year=2008}}, section 4.4. Another logarithmic scale is apparent magnitude. It measures the brightness of stars logarithmically.{{Citation|last1=Bradt|first1=Hale|title=Astronomy methods: a physical approach to astronomical observations|publisher=Cambridge University Press|series=Cambridge Planetary Science|isbn=978-0-521-53551-9|year=2004}}, section 8.3, p. 231 Yet another example is pH in chemistry; pH is the negative of the common logarithm of the activity of hydronium ions (the form hydrogen ions {{chem|H|+|}} take in water).{{Citation|author=IUPAC|title=Compendium of Chemical Terminology ("Gold Book")|edition=2nd|editor=A. D. McNaught, A. Wilkinson|publisher=Blackwell Scientific Publications|location=Oxford|year=1997|url=http://goldbook.iupac.org/P04524.html|isbn=978-0-9678550-9-7|doi=10.1351/goldbook}} The activity of hydronium ions in neutral water is 10âˆ’7 molÂ·Lâˆ’1, hence a pH of 7. Vinegar typically has a pH of about 3. The difference of 4 corresponds to a ratio of 104 of the activity, that is, vinegar's hydronium ion activity is about {{math|10âˆ’3 molÂ·Lâˆ’1}}.Semilog (log-linear) graphs use the logarithmic scale concept for visualization: one axis, typically the vertical one, is scaled logarithmically. For example, the chart at the right compresses the steep increase from 1 million to 1 trillion to the same space (on the vertical axis) as the increase from 1 to 1 million. In such graphs, exponential functions of the form {{math|1=f(x) = a Â· {{mvar|b}}x}} appear as straight lines with slope equal to the logarithm of {{mvar|b}}.Log-log graphs scale both axes logarithmically, which causes functions of the form {{math|1=f(x) = a Â· {{mvar|x}}k}} to be depicted as straight lines with slope equal to the exponent k. This is applied in visualizing and analyzing power laws.{{Citation|last1=Bird|first1=J.O.|title=Newnes engineering mathematics pocket book |publisher=Newnes|location=Oxford|edition=3rd|isbn=978-0-7506-4992-6|year=2001}}, section 34Psychology
Logarithms occur in several laws describing human perception:{{Citation | last1=Goldstein | first1=E. Bruce | title=Encyclopedia of Perception | url={{google books |plainurl=y |id=Y4TOEN4f5ZMC}} | publisher=Sage | location=Thousand Oaks, CA | series=Encyclopedia of Perception | isbn=978-1-4129-4081-8 | year=2009}}, pp. 355â€“56{{Citation | last1=Matthews | first1=Gerald | title=Human performance: cognition, stress, and individual differences | url={{google books |plainurl=y |id=0XrpulSM1HUC}} | publisher=Psychology Press | location=Hove | series=Human Performance: Cognition, Stress, and Individual Differences | isbn=978-0-415-04406-6 | year=2000}}, p. 48Hick's law proposes a logarithmic relation between the time individuals take to choose an alternative and the number of choices they have.{{Citation|last1=Welford|first1=A.T.|title=Fundamentals of skill|publisher=Methuen|location=London|isbn=978-0-416-03000-6 |oclc=219156|year=1968}}, p. 61 Fitts's law predicts that the time required to rapidly move to a target area is a logarithmic function of the distance to and the size of the target.{{Citation|author=Paul M. Fitts|date=June 1954|title=The information capacity of the human motor system in controlling the amplitude of movement|journal=Journal of Experimental Psychology|volume=47|issue=6|pages=381â€“91 | pmid=13174710 | doi =10.1037/h0055392 }}, reprinted in {{Citation|journal=Journal of Experimental Psychology: General|volume=121|issue=3|pages=262â€“69|year=1992 | pmid=1402698 | url=http://sing.stanford.edu/cs303-sp10/papers/1954-Fitts.pdf | format=PDF | accessdate=30 March 2011 |title=The information capacity of the human motor system in controlling the amplitude of movement|author=Paul M. Fitts|doi=10.1037/0096-3445.121.3.262}} In psychophysics, the Weberâ€“Fechner law proposes a logarithmic relationship between stimulus and sensation such as the actual vs. the perceived weight of an item a person is carrying.{{Citation | last1=Banerjee | first1=J.C. | title=Encyclopaedic dictionary of psychological terms | publisher=M.D. Publications | location=New Delhi | isbn=978-81-85880-28-0 | oclc=33860167 | year=1994|url={{google books |plainurl=y |id=Pwl5U2q5hfcC|page=306}} |page=304}} (This "law", however, is less realistic than more recent models, such as Stevens's power law.{{Citation|last1=Nadel|first1=Lynn|author1-link=Lynn Nadel|title=Encyclopedia of cognitive science|publisher=John Wiley & Sons|location=New York|isbn=978-0-470-01619-0|year=2005}}, lemmas Psychophysics and Perception: Overview)Psychological studies found that individuals with little mathematics education tend to estimate quantities logarithmically, that is, they position a number on an unmarked line according to its logarithm, so that 10 is positioned as close to 100 as 100 is to 1000. Increasing education shifts this to a linear estimate (positioning 1000 10x as far away) in some circumstances, while logarithms are used when the numbers to be plotted are difficult to plot linearly.{{Citation|doi=10.1111/1467-9280.02438|last1=Siegler|first1=Robert S.|last2=Opfer|first2=John E.|title=The Development of Numerical Estimation. Evidence for Multiple Representations of Numerical Quantity|volume=14|issue=3|pages=237â€“43|year=2003|journal=Psychological Science|url=http://www.psy.cmu.edu/~siegler/sieglerbooth-cd04.pdf|pmid=12741747|citeseerx=10.1.1.727.3696|access-date=7 January 2011|archive-url=https://web.archive.org/web/20110517002232weblink|archive-date=17 May 2011|dead-url=yes|df=dmy-all}}{{Citation|last1=Dehaene| first1=Stanislas|last2=Izard|first2=VÃ©ronique |last3=Spelke| first3=Elizabeth|last4=Pica| first4=Pierre| title=Log or Linear? Distinct Intuitions of the Number Scale in Western and Amazonian Indigene Cultures|volume=320|issue=5880|pages=1217â€“20|doi=10.1126/science.1156540|pmc=2610411|pmid=18511690| year=2008|journal=Science|postscript=|bibcode=2008Sci...320.1217D| citeseerx=10.1.1.362.2390}}Probability theory and statistics
File:PDF-log normal distributions.svg|thumb|right|alt=Three asymmetric PDF curves|Three Î¼}}, which is zero for all three of the PDFs shown, is the mean of the logarithm of the random variable, not the mean of the variable itself.File:Benfords law illustrated by world's countries population.png|Distribution of first digits (in %, red bars) in the population of the 237 countriespopulation of the 237 countriesLogarithms arise in probability theory: the law of large numbers dictates that, for a fair coin, as the number of coin-tosses increases to infinity, the observed proportion of heads approaches one-half. The fluctuations of this proportion about one-half are described by the law of the iterated logarithm.{{Citation | last1=Breiman | first1=Leo | title=Probability | publisher=Society for Industrial and Applied Mathematics | location=Philadelphia | series=Classics in applied mathematics | isbn=978-0-89871-296-4 | year=1992}}, section 12.9Logarithms also occur in log-normal distributions. When the logarithm of a random variable has a normal distribution, the variable is said to have a log-normal distribution.{{Citation|last1=Aitchison|first1=J.|last2=Brown|first2=J.A.C.|title=The lognormal distribution|publisher=Cambridge University Press|isbn=978-0-521-04011-2 |oclc=301100935|year=1969}} Log-normal distributions are encountered in many fields, wherever a variable is formed as the product of many independent positive random variables, for example in the study of turbulence.{hide}Citation
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Logarithms are used for maximum-likelihood estimation of parametric statistical models. For such a model, the likelihood function depends on at least one parameter that must be estimated. A maximum of the likelihood function occurs at the same parameter-value as a maximum of the logarithm of the likelihood (the "log likelihood"), because the logarithm is an increasing function. The log-likelihood is easier to maximize, especially for the multiplied likelihoods for independent random variables.{{Citation|last1=Rose|first1=Colin|last2=Smith|first2=Murray D.|title=Mathematical statistics with Mathematica|publisher=Springer-Verlag|location=Berlin, New York|series=Springer texts in statistics|isbn=978-0-387-95234-5|year=2002}}, section 11.3Benford's law describes the occurrence of digits in many data sets, such as heights of buildings. According to Benford's law, the probability that the first decimal-digit of an item in the data sample is d (from 1 to 9) equals {{math|log10(d + 1) âˆ’ log10(d)}}, regardless of the unit of measurement.{{Citation|last1=Tabachnikov|first1=Serge|authorlink1=Sergei Tabachnikov|title=Geometry and Billiards|publisher=American Mathematical Society|location=Providence, RI|isbn=978-0-8218-3919-5|year=2005|pages=36â€“40}}, section 2.1 Thus, about 30% of the data can be expected to have 1 as first digit, 18% start with 2, etc. Auditors examine deviations from Benford's law to detect fraudulent accounting.JOURNAL, The Effective Use of Benford's Law in Detecting Fraud in Accounting Data, Cindy, Durtschi, William, Hillison, Carl, Pacini,weblink V, 17â€“34, 2004, Journal of Forensic Accounting,weblink" title="web.archive.org/web/20170829062510weblink">weblink 29 August 2017, 28 May 2018, | author = Jean Mathieu and Julian Scott
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Computational complexity
Analysis of algorithms is a branch of computer science that studies the performance of algorithms (computer programs solving a certain problem).{{Citation|last1=Wegener|first1=Ingo| title=Complexity theory: exploring the limits of efficient algorithms|publisher=Springer-Verlag|location=Berlin, New York|isbn=978-3-540-21045-0|year=2005}}, pp. 1â€“2 Logarithms are valuable for describing algorithms that divide a problem into smaller ones, and join the solutions of the subproblems.{{Citation|last1=Harel|first1=David|last2=Feldman|first2=Yishai A.|title=Algorithmics: the spirit of computing|location=New York|publisher=Addison-Wesley|isbn=978-0-321-11784-7|year=2004}}, p. 143For example, to find a number in a sorted list, the binary search algorithm checks the middle entry and proceeds with the half before or after the middle entry if the number is still not found. This algorithm requires, on average, {{math|log2(N)}} comparisons, where N is the list's length.{{citation | last = Knuth | first = Donald | authorlink = Donald Knuth | title = The Art of Computer Programming | publisher = Addison-Wesley |location=Reading, MA | year= 1998| isbn = 978-0-201-89685-5 | title-link = The Art of Computer Programming }}, section 6.2.1, pp. 409â€“26 Similarly, the merge sort algorithm sorts an unsorted list by dividing the list into halves and sorting these first before merging the results. Merge sort algorithms typically require a time approximately proportional to {{math|N Â· log(N)}}.{{Harvard citations|last = Knuth | first = Donald|year=1998|loc=section 5.2.4, pp. 158â€“68|nb=yes}} The base of the logarithm is not specified here, because the result only changes by a constant factor when another base is used. A constant factor is usually disregarded in the analysis of algorithms under the standard uniform cost model.{{Citation|last1=Wegener|first1=Ingo| title=Complexity theory: exploring the limits of efficient algorithms|publisher=Springer-Verlag|location=Berlin, New York|isbn=978-3-540-21045-0|year=2005|page=20}}A function {{math|f(x)}} is said to grow logarithmically if {{math|f(x)}} is (exactly or approximately) proportional to the logarithm of {{mvar|x}}. (Biological descriptions of organism growth, however, use this term for an exponential function.{{Citation |last1=Mohr|first1=Hans|last2=Schopfer|first2=Peter|title=Plant physiology|publisher=Springer-Verlag|location=Berlin, New York|isbn=978-3-540-58016-4|year=1995}}, chapter 19, p. 298) For example, any natural number N can be represented in binary form in no more than {{math|log2(N) + 1}} bits. In other words, the amount of memory needed to store N grows logarithmically with N.Entropy and chaos
File:Chaotic Bunimovich stadium.png|right|thumb|Billiards on an oval billiard table. Two particles, starting at the center with an angle differing by one degree, take paths that diverge chaotically because of reflectionsreflectionsEntropy is broadly a measure of the disorder of some system. In statistical thermodynamics, the entropy S of some physical system is defined as
S = - k sum_i p_i ln(p_i).,
The sum is over all possible states i of the system in question, such as the positions of gas particles in a container. Moreover, {{math|pi}} is the probability that the state i is attained and k is the Boltzmann constant. Similarly, entropy in information theory measures the quantity of information. If a message recipient may expect any one of N possible messages with equal likelihood, then the amount of information conveyed by any one such message is quantified as {{math|log2(N)}} bits.{{Citation|last1=Eco|first1=Umberto|author1-link=Umberto Eco|title=The open work |publisher=Harvard University Press|isbn=978-0-674-63976-8|year=1989}}, section III.ILyapunov exponents use logarithms to gauge the degree of chaoticity of a dynamical system. For example, for a particle moving on an oval billiard table, even small changes of the initial conditions result in very different paths of the particle. Such systems are chaotic in a deterministic way, because small measurement errors of the initial state predictably lead to largely different final states.{{Citation | last1=Sprott | first1=Julien Clinton | title=Elegant Chaos: Algebraically Simple Chaotic Flows | journal=Elegant Chaos: Algebraically Simple Chaotic Flows. Edited by Sprott Julien Clinton. Published by World Scientific Publishing Co. Pte. Ltd | url={{google books |plainurl=y |id=buILBDre9S4C}} | publisher=World Scientific |location=New Jersey|isbn=978-981-283-881-0| year=2010| bibcode=2010ecas.book.....S | doi=10.1142/7183 }}, section 1.9 At least one Lyapunov exponent of a deterministically chaotic system is positive.Fractals
File:Sierpinski dimension.svg|The Sierpinski triangle (at the right) is constructed by repeatedly replacing right|thumb|400px|alt=Parts of a triangle are removed in an iterated way.Logarithms occur in definitions of the dimension of fractals.{{Citation|last1=Helmberg|first1=Gilbert|title=Getting acquainted with fractals|publisher=Walter de Gruyter|series=De Gruyter Textbook|location=Berlin, New York|isbn=978-3-11-019092-2|year=2007}} Fractals are geometric objects that are self-similar: small parts reproduce, at least roughly, the entire global structure. The Sierpinski triangle (pictured) can be covered by three copies of itself, each having sides half the original length. This makes the Hausdorff dimension of this structure {{math|1=ln(3)/ln(2) â‰ˆ 1.58}}. Another logarithm-based notion of dimension is obtained by counting the number of boxes needed to cover the fractal in question.Music
{{multiple image| direction = vertical| width = 350| footer = Four different octaves shown on a linear scale, then shown on a logarithmic scale (as the ear hears them).| image1 = 4octavesAndfrequencies.jpg| alt1 = Four different octaves shown on a linear scale.| image2 = 4octavesAndfrequenciesEars.jpg| alt2 = Four different octaves shown on a logarithmic scale.}}Logarithms are related to musical tones and intervals. In equal temperament, the frequency ratio depends only on the interval between two tones, not on the specific frequency, or pitch, of the individual tones. For example, the note A has a frequency of 440 Hz and B-flat has a frequency of 466 Hz. The interval between A and B-flat is a semitone, as is the one between B-flat and B (frequency 493 Hz). Accordingly, the frequency ratios agree:
frac{466}{440} approx frac{493}{466} approx 1.059 approx sqrt[12]2.
Therefore, logarithms can be used to describe the intervals: an interval is measured in semitones by taking the {{math|base-21/12}} logarithm of the frequency ratio, while the {{math|base-21/1200}} logarithm of the frequency ratio expresses the interval in cents, hundredths of a semitone. The latter is used for finer encoding, as it is needed for non-equal temperaments.{{Citation|last1=Wright|first1=David|title=Mathematics and music|location=Providence, RI|publisher=AMS Bookstore|isbn=978-0-8218-4873-9|year=2009}}, chapter 5{| class="wikitable" style="text-align:center; margin:1em auto 1em auto;"Interval(the two tones are played at the same time) | 72 tone equal temperament | {{audio>1_step_in_72-et_on_C.mid|play}} | Semitone {{audio | Minor_second_on_C.mid|play}} | Just major third {{audio | Just_major_third_on_C.mid|play}} | Major third {{audio | Major_third_on_C.mid|play}} | Tritone {{audio | Tritone_on_C.mid|play}} | Octave {{audio | Perfect_octave_on_C.mid|play}} |
Frequency ratio r | 2^{frac 1 {72}} approx 1.0097 | 2^{frac 1 {12}} approx 1.0595 | tfrac 5 4 = 1.25 | begin{align} 2^{frac 4 {12}} & = sqrt[3] 2 & approx 1.2599 end{align} | begin{align} 2^{frac 6 {12}} & = sqrt 2 & approx 1.4142 end{align} | 2^{frac {12} {12}} = 2 |
Corresponding number of semitoneslog_{sqrt[12] 2}(r) = 12 log_2 (r) | tfrac 1 6 , | 1 , | approx 3.8631 , | 4 , | 6 , | 12 , |
Corresponding number of centslog_{sqrt[1200] 2}(r) = 1200 log_2 (r) | 16 tfrac 2 3 , | 100 , | approx 386.31 , | 400 , | 600 , | 1200 , |
Number theory
Natural logarithms are closely linked to counting prime numbers (2, 3, 5, 7, 11, ...), an important topic in number theory. For any integer {{mvar|x}}, the quantity of prime numbers less than or equal to {{mvar|x}} is denoted {{math|{{pi}}(x)}}. The prime number theorem asserts that {{math|{{pi}}(x)}} is approximately given by
frac{x}{ln(x)},
in the sense that the ratio of {{math|{{pi}}(x)}} and that fraction approaches 1 when {{mvar|x}} tends to infinity.{{Citation|last1=Bateman|first1=P.T.|last2=Diamond|first2=Harold G.|title=Analytic number theory: an introductory course|publisher=World Scientific|location=New Jersey|isbn=978-981-256-080-3 |oclc=492669517|year=2004}}, theorem 4.1 As a consequence, the probability that a randomly chosen number between 1 and {{mvar|x}} is prime is inversely proportional to the number of decimal digits of {{mvar|x}}. A far better estimate of {{math|{{pi}}(x)}} is given by theoffset logarithmic integral function {{math|Li(x)}}, defined by
mathrm{Li}(x) = int_2^x frac1{ln(t)} ,dt.
The Riemann hypothesis, one of the oldest open mathematical conjectures, can be stated in terms of comparing {{math|{{pi}}(x)}} and {{math|Li(x)}}.{{Harvard citations|last1=Bateman|first1=P. T.|last2=Diamond|year=2004|nb=yes |loc=Theorem 8.15}} The ErdÅ‘sâ€“Kac theorem describing the number of distinct prime factors also involves the natural logarithm.The logarithm of n factorial, {{math|1=n! = 1 Â· 2 Â· ... Â· n}}, is given by
ln (n!) = ln (1) + ln (2) + cdots + ln (n). ,
This can be used to obtain Stirling's formula, an approximation of {{math|n!}} for large n.{{Citation|last1=Slomson|first1=Alan B.|title=An introduction to combinatorics|publisher=CRC Press|location=London|isbn=978-0-412-35370-3|year=1991}}, chapter 4Generalizations
Complex logarithm
(File:Complex number illustration multiple arguments.svg|thumb|right|Polar form of {{math|z {{=}} x + iy}}. Both {{mvar|Ï†}} and {{mvar|Ï†'}} are arguments of {{mvar|z}}.|alt=An illustration of the polar form: a point is described by an arrow or equivalently by its length and angle to the {{mvar|x}} axis.)All the complex numbers {{mvar|a}} that solve the equation
e^a=z
are called complex logarithms of {{mvar|z}}, when {{mvar|z}} is (considered as) a complex number. A complex number is commonly represented as {{math|z {{=}} x + iy}}, where {{mvar|x}} and {{mvar|y}} are real numbers and {{mvar|i}} is an imaginary unit, the square of which is âˆ’1. Such a number can be visualized by a point in the complex plane, as shown at the right. The polar form encodes a non-zero complex number {{mvar|z}} by its absolute value, that is, the (positive, real) distance {{math|r}} to the origin, and an angle between the real ({{mvar|x}}) axis Re and the line passing through both the origin and {{mvar|z}}. This angle is called the argument of {{mvar|z}}.The absolute value {{mvar|r}} of {{mvar|z}} is given by
textstyle r=sqrt{x^2+y^2}.
Using the geometrical interpretation of sin and cos and their periodicity in 2pi, any complex number {{mvar|z}} may be denoted as
z = x + iy = r (cos varphi + i sin varphi )= r (cos (varphi + 2kpi) + i sin (varphi + 2kpi)),
for any integer number {{mvar|k}}. Evidently the argument of {{mvar|z}} is not uniquely specified: both {{mvar|Ï†}} and {{mvar|Ï†}}' = {{mvar|Ï†}} + 2k{{pi}} are valid arguments of {{mvar|z}} for all integers {{mvar|k}}, because adding 2k{{pi}} radian or kâ‹…360Â°{{refn|See radian for the conversion between 2{{pi}} and 360 degree.|group=nb}} to {{mvar|Ï†}} corresponds to "winding" around the origin counter-clock-wise by {{mvar|k}} turns. The resulting complex number is always {{mvar|z}}, as illustrated at the right for {{math|k {{=}} 1}}. One may select exactly one of the possible arguments of {{mvar|z}} as the so-called principal argument, denoted {{math|Arg(z)}}, with a capital {{math|A}}, by requiring {{mvar|Ï†}} to belong to one, conveniently selected turn, e.g., -pi < varphi le pi{{Citation|last1=Ganguly|location=Kolkata|first1=S.|title=Elements of Complex Analysis|publisher=Academic Publishers|isbn=978-81-87504-86-3|year=2005}}, Definition 1.6.3 or 0 le varphi < 2pi.{{Citation|last1=Nevanlinna|first1=Rolf Herman|author1-link=Rolf Nevanlinna|last2=Paatero|first2=Veikko|title=Introduction to complex analysis|journal=London: Hilger|location=Providence, RI|publisher=AMS Bookstore|isbn=978-0-8218-4399-4|year=2007|bibcode=1974aitc.book.....W}}, section 5.9 These regions, where the argument of {{mvar|z}} is uniquely determined are called branches of the argument function.(File:Complex log.jpg|right|thumb|The principal branch (-{{pi}}, {{pi}}) of the complex logarithm, {{math|Log(z)}}. The black point at {{math|z {{=}} 1}} corresponds to absolute value zero and brighter (more saturated) colors refer to bigger absolute values. The Log(z)}}.|alt=A density plot. In the middle there is a black point, at the negative axis the hue jumps sharply and evolves smoothly otherwise.Euler's formula connects the trigonometric functions sine and cosine to the complex exponential:
e^{ivarphi} = cos varphi + isin varphi .
Using this formula, and again the periodicity, the following identities hold:{{Citation|last1=Moore|first1=Theral Orvis|last2=Hadlock|first2=Edwin H.|title=Complex analysis|publisher=World Scientific|location=Singapore|isbn=978-981-02-0246-0|year=1991}}, section 1.2
begin{array}{lll}z& = & r left (cos varphi + i sin varphiright)
& = & r left (cos(varphi + 2kpi) + i sin(varphi + 2kpi)right) & = & r e^{i (varphi + 2kpi)} & = & e^{ln(r)} e^{i (varphi + 2kpi)} & = & e^{ln(r) + i(varphi + 2kpi)} = e^{a_k},end{array}where {{math|ln(r)}} is the unique real natural logarithm, {{math|a'k}} denote the complex logarithms of {{mvar|z}}, and {{mvar|k}} is an arbitrary integer. Therefore, the complex logarithms of {{mvar|z}}, which are all those complex values {{math|a'k}} for which the {{math|ak-th}} power of {{mvar|e}} equals {{mvar|z}}, are the infinitely many values
a_k = ln (r) + i ( varphi + 2 k pi ),quad for arbitrary integers {{mvar|k}}.
Taking {{mvar|k}} such that varphi + 2 k pi is within the defined interval for the principal arguments, then {{math|ak}} is called the principal value of the logarithm, denoted {{math|Log(z)}}, again with a capital {{math|L}}. The principal argument of any positive real number {{mvar|x}} is 0; hence {{math|Log(x)}} is a real number and equals the real (natural) logarithm. However, the above formulas for logarithms of products and powers do not generalize to the principal value of the complex logarithm.{{Citation | last1=Wilde | first1=Ivan Francis | title=Lecture notes on complex analysis | publisher=Imperial College Press | location=London | isbn=978-1-86094-642-4 | year=2006|url=https://books.google.com/?id=vrWES2W6vG0C&pg=PA97&dq=complex+logarithm#v=onepage&q=complex%20logarithm&f=false}}, theorem 6.1.The illustration at the right depicts {{math|Log(z)}}, confining the arguments of {{mvar|z}} to the interval {{math|(-{{pi}}, {{pi}}]}}. This way the corresponding branch of the complex logarithm has discontinuities all along the negative real {{mvar|x}} axis, which can be seen in the jump in the hue there. This discontinuity arises from jumping to the other boundary in the same branch, when crossing a boundary, i.e., not changing to the corresponding {{mvar|k}}-value of the continuously neighboring branch. Such a locus is called a branch cut. Dropping the range restrictions on the argument makes the relations "argument of {{mvar|z}}", and consequently the "logarithm of {{mvar|z}}", multi-valued functions.Inverses of other exponential functions
Exponentiation occurs in many areas of mathematics and its inverse function is often referred to as the logarithm. For example, the logarithm of a matrix is the (multi-valued) inverse function of the matrix exponential.{{Citation|last1=Higham|first1=Nicholas|author1-link=Nicholas Higham|title=Functions of Matrices. Theory and Computation|location=Philadelphia, PA|publisher=SIAM|isbn=978-0-89871-646-7|year=2008}}, chapter 11. Another example is the p-adic logarithm, the inverse function of the p-adic exponential. Both are defined via Taylor series analogous to the real case.{{Neukirch ANT}}, section II.5. In the context of differential geometry, the exponential map maps the tangent space at a point of a manifold to a neighborhood of that point. Its inverse is also called the logarithmic (or log) map.{{Citation|last1=Hancock|first1=Edwin R.|last2=Martin|first2=Ralph R.|last3=Sabin|first3=Malcolm A.|title=Mathematics of Surfaces XIII: 13th IMA International Conference York, UK, September 7â€“9, 2009 Proceedings|url=https://books.google.com/books?id=0cqCy9x7V_QC&pg=PA379|publisher=Springer|year=2009|page=379|isbn=978-3-642-03595-1}}In the context of finite groups exponentiation is given by repeatedly multiplying one group element {{mvar|b}} with itself. The discrete logarithm is the integer n solving the equation
b^n = x,,
where {{mvar|x}} is an element of the group. Carrying out the exponentiation can be done efficiently, but the discrete logarithm is believed to be very hard to calculate in some groups. This asymmetry has important applications in public key cryptography, such as for example in the Diffieâ€“Hellman key exchange, a routine that allows secure exchanges of cryptographic keys over unsecured information channels.{{Citation|last1=Stinson|first1=Douglas Robert|title=Cryptography: Theory and Practice|publisher=CRC Press|location=London|edition=3rd|isbn=978-1-58488-508-5|year=2006}} Zech's logarithm is related to the discrete logarithm in the multiplicative group of non-zero elements of a finite field.{{Citation|last1=Lidl|first1=Rudolf|last2=Niederreiter|first2=Harald|author2-link = Harald Niederreiter |title=Finite fields|publisher=Cambridge University Press|isbn=978-0-521-39231-0|year=1997}}{{anchor|double logarithm}}Further logarithm-like inverse functions include the double logarithm ln(ln(x)), the super- or hyper-4-logarithm (a slight variation of which is called iterated logarithm in computer science), the Lambert W function, and the logit. They are the inverse functions of the double exponential function, tetration, of {{math|f(w) {{=}} wew}},{{Citation | last1=Corless | first1=R. | last2=Gonnet | first2=G. | last3=Hare | first3=D. | last4=Jeffrey | first4=D. | last5=Knuth | first5=Donald | author5-link=Donald Knuth | title=On the Lambert W function | url=http://www.apmaths.uwo.ca/~djeffrey/Offprints/W-adv-cm.pdf | year=1996 | journal=Advances in Computational Mathematics | issn=1019-7168 | volume=5 | pages=329â€“59 | doi=10.1007/BF02124750 | access-date=13 February 2011 | archive-url=https://web.archive.org/web/20101214110615weblink | archive-date=14 December 2010 | dead-url=yes | df=dmy-all }} and of the logistic function, respectively.{{Citation | last1=Cherkassky | first1=Vladimir | last2=Cherkassky | first2=Vladimir S. | last3=Mulier | first3=Filip | title=Learning from data: concepts, theory, and methods | publisher=John Wiley & Sons | location=New York | series=Wiley series on adaptive and learning systems for signal processing, communications, and control | isbn=978-0-471-68182-3 | year=2007}}, p. 357Related concepts
From the perspective of group theory, the identity {{math|log(cd) {{=}} log(c) + log(d)}} expresses a group isomorphism between positive reals under multiplication and reals under addition. Logarithmic functions are the only continuous isomorphisms between these groups.{{Citation|last1=Bourbaki|first1=Nicolas|author1-link=Nicolas Bourbaki|title=General topology. Chapters 5â€“10|publisher=Springer-Verlag|location=Berlin, New York|series=Elements of Mathematics|isbn=978-3-540-64563-4|mr=1726872|year=1998}}, section V.4.1 By means of that isomorphism, the Haar measure (Lebesgue measure) dx on the reals corresponds to the Haar measure {{math|dx/x}} on the positive reals.{{Citation|last1=Ambartzumian|first1=R.V.|authorlink=Rouben V. Ambartzumian|title=Factorization calculus and geometric probability|publisher=Cambridge University Press|isbn=978-0-521-34535-4|year=1990}}, section 1.4 The non-negative reals not only have a multiplication, but also have addition, and form a semiring, called the probability semiring; this is in fact a semifield. The logarithm then takes multiplication to addition (log multiplication), and takes addition to log addition (LogSumExp), giving an isomorphism of semirings between the probability semiring and the log semiring.Logarithmic one-forms {{math|df/f}} appear in complex analysis and algebraic geometry as differential forms with logarithmic poles.{{Citation|last1=Esnault|first1=HÃ©lÃ¨ne|last2=Viehweg|first2=Eckart|title=Lectures on vanishing theorems|location=Basel, Boston|publisher=BirkhÃ¤user Verlag|series=DMV Seminar|isbn=978-3-7643-2822-1|mr=1193913|year=1992|volume=20|doi=10.1007/978-3-0348-8600-0|citeseerx=10.1.1.178.3227}}, section 2The polylogarithm is the function defined bySee also
Notes
{{reflist|group=nb|30em}}References
{{Reflist}}External links
- {{Commons category inline}}
- {{Wiktionary-inline}}
- weblink" title="web.archive.org/web/20121218200616weblink">Khan Academy: Logarithms, free online micro lectures
- {{springer|title=Logarithmic function|id=p/l060600}}
- {{Citation|author=Colin Byfleet|url=http://mediasite.oddl.fsu.edu/mediasite/Viewer/?peid=003298f9a02f468c8351c50488d6c479|title=Educational video on logarithms|accessdate=2010-10-12}}
- {{Citation|author=Edward Wright |url=http://www.johnnapier.com/table_of_logarithms_001.htm |title=Translation of Napier's work on logarithms |accessdate=2010-10-12 |deadurl=unfit |archiveurl=https://web.archive.org/web/20021203005508weblink |archivedate=3 December 2002 }}
- EB1911, Logarithm, 16, 868â€“77, James Whitbread Lee, Glaisher,
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