Editing Binary logarithm (section)

==Notation==
In mathematics, the binary logarithm of a number {{mvar|n}} is often written as {{math|log<sub>2</sub>{{hsp}}''n''}}.<ref>For instance, this is the notation used in the ''[[Encyclopedia of Mathematics]]'' and ''[[The Princeton Companion to Mathematics]]''.</ref> However, several other notations for this function have been used or proposed, especially in application areas.

Some authors write the binary logarithm as {{math|lg ''n''}},<ref name="clrs">{{citation 
| last1 = Cormen | first1 = Thomas H. | author1-link = Thomas H. Cormen
| last2 = Leiserson | first2 = Charles E. | author2-link = Charles E. Leiserson
| last3 = Rivest | first3 = Ronald L. | author3-link = Ron Rivest
| last4 = Stein | first4 = Clifford | author4-link = Clifford Stein
| title = Introduction to Algorithms
| orig-year = 1990
| year = 2001
| edition = 2nd
| publisher = MIT Press and McGraw-Hill
| isbn = 0-262-03293-7
| pages = 34, 53–54
| title-link = Introduction to Algorithms }}</ref><ref name="sw11">{{citation
 | title=Algorithms
 | first1=Robert | last1=Sedgewick | author1-link=Robert Sedgewick (computer scientist)
 | first2=Kevin Daniel | last2=Wayne
 | publisher=Addison-Wesley Professional
 | year=2011
 | isbn=978-0-321-57351-3
 | page=185
 |url = https://books.google.com/books?id=MTpsAQAAQBAJ&pg=PA185
}}.</ref> the notation listed in ''[[The Chicago Manual of Style]]''.<ref>{{citation
 | title=The Chicago Manual of Style
 | year=2003
 | edition=25th
 | publisher=University of Chicago Press
 | page=530
| title-link=The Chicago Manual of Style
 }}.</ref> [[Donald Knuth]] credits this notation to a suggestion of [[Edward Reingold]],<ref name="knuth">{{citation
 | series=[[The Art of Computer Programming]]
 | volume=1
 | title=Fundamental Algorithms
 | first=Donald E. | last=Knuth | author-link=Donald Knuth
 | edition=3rd
 | publisher=Addison-Wesley Professional
 | year=1997
 | isbn=978-0-321-63574-7}}, [https://books.google.com/books?id=x9AsAwAAQBAJ&pg=PA11 p.&nbsp;11]. The same notation was in the 1973 2nd edition of the same book (p.&nbsp;23) but without the credit to Reingold.</ref> but its use in both information theory and computer science dates to before Reingold was active.<ref>{{citation
 | last = Trucco | first = Ernesto
 | doi = 10.1007/BF02477836
 | journal = Bull. Math. Biophys.
 | mr = 0077919
 | pages = 129–135
 | title = A note on the information content of graphs
 | volume = 18
 | issue = 2
 | year = 1956
}}.</ref><ref name=mitchell>{{citation
 | last = Mitchell | first = John N.
 | doi = 10.1109/TEC.1962.5219391
 | issue = 4
 | journal = IRE Transactions on Electronic Computers
 | pages = 512–517
 | title = Computer multiplication and division using binary logarithms
 | volume = EC-11
 | year = 1962
}}.</ref>  The binary logarithm has also been written as {{math|log ''n''}} with a prior statement that the default base for the logarithm is&nbsp;{{math|2}}.<ref>{{citation
 | title=Mathematics for Engineers
 | first1=Georges | last1=Fiche
 | first2=Gerard | last2=Hebuterne
 | publisher=John Wiley & Sons
 | year=2013
 | isbn=978-1-118-62333-6
 | page=152
 | url=https://books.google.com/books?id=TqkckiuuXg8C&pg=PT152
 | quote=In the following, and unless otherwise stated, the notation {{math|log ''x''}} always stands for the logarithm to the base {{math|2}} of {{mvar|x}}
}}.</ref><ref>{{citation
 | title=Elements of Information Theory
 | first1=Thomas M. | last1=Cover | author1-link = Thomas M. Cover
 | first2=Joy A. | last2=Thomas
 | edition=2nd
 | publisher=[[John Wiley & Sons]]
 | year=2012
 | isbn=978-1-118-58577-1
 | page=33
 | url=https://books.google.com/books?id=VWq5GG6ycxMC&pg=PT33
 | quote=Unless otherwise specified, we will take all logarithms to base {{math|2}}
}}.</ref><ref name="gt02">{{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&nbsp;... As is the custom in the computing literature, we omit writing the base {{mvar|b}} of the logarithm when {{math|1=''b''&nbsp;=&nbsp;2}}.
}}</ref> Another notation that is often used for the same function (especially in the German scientific literature) is {{math|ld ''n''}},<ref name="Tafel_1971">{{citation |title=Einführung in die digitale Datenverarbeitung |language=de |trans-title=Introduction to digital information processing |first=Hans Jörg |last=Tafel |publisher=[[Carl Hanser Verlag]] |date=1971 |location=Munich |isbn=3-446-10569-7 |pages=20–21}}</ref><ref name="Tietze_Schenk_1999">{{citation |title=Halbleiter-Schaltungstechnik |url=https://archive.org/details/halbleiterschalt00tiet_103 |url-access=limited |language=de |author-first1=Ulrich |author-last1=Tietze |author-first2=Christoph |author-last2=Schenk |edition=1st corrected reprint, 11th |date=1999 |publisher=[[Springer Verlag]] |isbn=3-540-64192-0 |page=[https://archive.org/details/halbleiterschalt00tiet_103/page/n1395 1370]}}</ref><ref name="Bauer_2009">{{citation
 | title=Origins and Foundations of Computing: In Cooperation with Heinz Nixdorf MuseumsForum
 | first=Friedrich L. | last=Bauer
 | publisher=[[Springer Science & Business Media]]
 | year=2009
 | isbn=978-3-642-02991-2
 | page=54
 | url=https://books.google.com/books?id=y4uTaLiN-wQC&pg=PA54
}}.</ref> from [[Latin]] ''[[wikt:en:logarithmus#Latin|logarithmus]] [[wikt:en:dualis#Latin|dualis]]''<ref name="Tafel_1971"/> or ''logarithmus dyadis''.<ref name="Tafel_1971"/>
The {{Interlanguage link multi|DIN 1302|de}}, [[ISO 31-11]] and [[ISO 80000-2]] standards recommend yet another notation, {{math|lb ''n''}}. According to these standards, {{math|lg ''n''}} should not be used for the binary logarithm, as it is instead reserved for the [[common logarithm]] {{math|log<sub>10</sub> ''n''}}.<ref>For DIN 1302 see {{citation
 | title=Brockhaus Enzyklopädie in zwanzig Bänden | language=de |trans-title=Brockhaus Encyclopedia in Twenty Volumes
 | volume=11 | page=554
 | publisher=F.A. Brockhaus | location=Wiesbaden
 | isbn=978-3-7653-0000-4
 | year=1970
}}.</ref><ref>For ISO 31-11 see {{citation
 | last1 = Thompson | first1 = Ambler
 | last2 = Taylor | first2 = Barry M
 | date = March 2008
 | page = 33
 | publisher = [[NIST]]
 | title = Guide for the Use of the International System of Units (SI) — NIST Special Publication 811, 2008 Edition — Second Printing
 | url = http://physics.nist.gov/cuu/pdf/sp811.pdf}}.</ref><ref>For ISO 80000-2 see {{citation
 | chapter-url=http://www.ise.ncsu.edu/jwilson/files/mathsigns.pdf
 | title=International Standard ISO 80000-2
 | chapter=Quantities and units – Part 2: Mathematical signs and symbols to be used in the natural sciences and technology
 | edition=1st|date=December 1, 2009
 | at=Section 12, Exponential and logarithmic functions, p.&nbsp;18}}.</ref>