மடக்கை: திருத்தங்களுக்கு இடையிலான வேறுபாடு

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Booradleyp (பேச்சு | பங்களிப்புகள்)
வரிசை 92:
அதாவது தரப்பட்ட எண் ''x'' மற்றும் அதன் மடக்கை log<sub>''b''</sub>(''x'') தெரிந்த அடிமானம் b க்கு பின்வருமாறு தரப்படும்:
: <math> b = x^\frac{1}{\log_b(x)}.</math>
 
==குறிப்பிட்ட அடிமானங்கள்==
அடிமானங்களில் ''b'' = 10, ''b'' = ''e'']] ( ≈ 2.71828), and ''b' = 2 மூன்றும் குறிப்பிடத் தக்கவை. கணிதத்தில் அடிமானம் ''e'' அதிகம் பயன்பாடு கொண்டுள்ளது. அடிமானம் 10, தசம எண்மான முறையில் கணக்கீடுகளை எளிதாகச் செய்யப் பயன்படுகிறது<ref>{{Citation|last1=Downing|first1=Douglas|title=Algebra the Easy Way|series=Barron's Educational Series|location=Hauppauge, N.Y.|publisher=Barron's|isbn=978-0-7641-1972-9|year=2003}}, chapter 17, p. 275</ref>
:<math>\log_{10}(10 x) = \log_{10}(10) + \log_{10}(x) = 1 + \log_{10}(x).\ </math>
Thus, log<sub>10</sub>(''x'') is related to the number of [[decimal digit]]s of a positive integer ''x'': the number of digits is the smallest [[integer]] strictly bigger than log<sub>10</sub>(''x'').<ref>{{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</ref> For example, log<sub>10</sub>(1430) is approximately 3.15. The next integer is 4, which is the number of digits of 1430. The logarithm to base two is used in [[computer science]], where the [[binary numeral system|binary system]] is ubiquitous.
 
The following table lists common notations for logarithms to these bases and the fields where they are used. Many disciplines write log(''x'') instead of log<sub>''b''</sub>(''x''), when the intended base can be determined from the context. The notation <sup>''b''</sup>log(''x'') also occurs.<ref>{{Citation| url=http://www.mathe-online.at/mathint/lexikon/l.html |author1=Franz Embacher |author2=Petra Oberhuemer |title=Mathematisches Lexikon |publisher=mathe online: für Schule, Fachhochschule, Universität unde Selbststudium |accessdate=22/03/2011 |language=German}}</ref> The "ISO notation" column lists designations suggested by the [[International Organization for Standardization]] ([[ISO 31-11]]).<ref>{{Citation| title = Guide for the Use of the International System of Units (SI)|author = B. N. Taylor|publisher = US Department of Commerce|year = 1995|url = http://physics.nist.gov/Pubs/SP811/sec10.html#10.1.2}}</ref>
 
{| class="wikitable" style="text-align:center; margin:1em auto 1em auto;"
|-
! scope="col"|Base ''b''
! scope="col"|Name for log<sub>''b''</sub>(''x'')
! scope="col"|ISO notation
! scope="col"|Other notations
! scope="col"|Used in
|-
! scope="row"|2
| [[binary logarithm]]
| lb(''x'')<ref name=gullberg>{{Citation|title = Mathematics: from the birth of numbers.|author = Gullberg, Jan|location=New York|publisher = W. W. Norton & Co|year = 1997|isbn=978-0-393-04002-9}}</ref>
| ld(''x''), log(''x''), lg(''x'')
| computer science, [[information theory]], mathematics
|-
! scope="row"|''e''
| [[natural logarithm]]
| ln(''x''){{#tag:ref|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.<ref>
{{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
}}</ref>
The notation was invented by [[Irving Stringham]], a mathematician.<ref>
{{Citation
|title = Uniplanar algebra: being part I of a propædeutic to the higher mathematical analysis
|author = Irving Stringham
|publisher = The Berkeley Press
|year = 1893
|page = xiii
|url = http://books.google.com/?id=hPEKAQAAIAAJ&pg=PR13&dq=%22Irving+Stringham%22+In-natural-logarithm&q=
}}</ref><ref>
{{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 = http://books.google.com/?id=APJ7QeR_XPkC&pg=PA59&dq=%22Irving+Stringham%22+logarithm+ln&q=%22Irving%20Stringham%22%20logarithm%20ln
}}</ref>|name=adaa|group=nb}}
| log(''x'')<br>(in mathematics and many [[programming language]]s{{#tag:ref|For example [[C (programming language)|C]], [[Java (programming language)|Java]], [[Haskell (programming language)|Haskell]], and [[BASIC programming language|BASIC]].|group=nb}})
| mathematical analysis, physics, chemistry,<br>[[statistics]], [[economics]], and some engineering fields
|-
! scope="row"|10
| [[common logarithm]]
| lg(''x'')
| log(''x'')<br>(in engineering, biology, astronomy),
| various [[engineering]] fields (see [[decibel]] and see below), <br>logarithm [[Mathematical table|tables]], handheld [[Scientific calculator|calculators]]
|}
 
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