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[[File:Binary logarithm plot with ticks.svg|thumbnail|right|upright=1.35|Graph of {{math|log<sub>2</sub>{{hsp}}''x''}} as a function of a positive real number {{mvar|x}}]]

In [[mathematics]], the '''binary logarithm''' ({{math|log<sub>2</sub>{{hsp}}''n''}}) is the [[exponentiation|power]] to which the number {{math|2}} must be [[exponentiation|raised]] to obtain the value&nbsp;{{mvar|n}}. That is, for any real number {{mvar|x}},
:<math>x=\log_2 n \quad\Longleftrightarrow\quad 2^x=n.</math>
For example, the binary logarithm of {{math|1}} is {{math|0}}, the binary logarithm of {{math|2}} is {{math|1}}, the binary logarithm of {{math|4}} is&nbsp;{{math|2}}, and the binary logarithm of {{math|32}} is&nbsp;{{math|5}}.

The binary logarithm is the [[logarithm]] to the base {{math|2}}. The binary logarithm function is the [[inverse function]] of the [[power of two]] function. As well as {{math|log<sub>2</sub>}}, alternative notations for the binary logarithm include {{math|lg}}, {{math|ld}}, {{math|lb}} (the notation preferred by [[ISO 31-11]] and [[ISO 80000-2]]), and (with a prior statement that the default base is&nbsp;2) {{math|log}}.

Historically, the first application of binary logarithms was in [[music theory]], by [[Leonhard Euler]]: the binary logarithm of a frequency ratio of two musical tones gives the number of [[octave]]s by which the tones differ. Binary logarithms can be used to calculate the length of the representation of a number in the [[binary numeral system]], or the number of [[bit]]s needed to encode a message in [[information theory]]. In [[computer science]], they count the number of steps needed for [[binary search]] and related algorithms. Other areas
in which the binary logarithm is frequently used include [[combinatorics]], [[bioinformatics]], the design of sports [[tournament]]s, and [[photography]].

Binary logarithms are included in the standard [[C mathematical functions]] and other mathematical software packages.
The integer part of a binary logarithm can be found using the [[find first set]] operation on an integer value, or by looking up the exponent of a [[floating point]] value.
The fractional part of the logarithm can be calculated efficiently.

==History==
{{Main|History of logarithms}}
[[File:Leonhard Euler.jpg|thumb|upright=0.75|[[Leonhard Euler]] was the first to apply binary logarithms to [[music theory]], in 1739.]]
The [[power of two|powers of two]] have been known since antiquity; for instance, they appear in [[Euclid's Elements|Euclid's ''Elements'']], Props. IX.32 (on the [[factorization]] of powers of two) and IX.36 (half of the [[Euclid–Euler theorem]], on the structure of even [[perfect number]]s).
And the binary logarithm of a power of two is just its position in the ordered sequence of powers of two.
On this basis, [[Michael Stifel]] has been credited with publishing the first known table of binary logarithms in 1544. His book ''Arithmetica Integra'' contains several tables that show the [[integer]]s with their corresponding powers of two. Reversing the rows of these tables allow them to be interpreted as tables of binary logarithms.<ref>
{{Citation|title = Precalculus mathematics|first1 = Vivian Shaw|last1= Groza |first2= Susanne M. |last2=Shelley|publisher = Holt, Rinehart and Winston|location=New York|year=1972|isbn=978-0-03-077670-0|page = 182|url = https://books.google.com/books?id=yM_lSq1eJv8C&pg=PA182}}.</ref><ref>{{citation
 | last = Stifel | first = Michael | author-link = Michael Stifel
 | language = Latin
 | page = 31
 | title = Arithmetica integra
 | url = https://books.google.com/books?id=fndPsRv08R0C&pg=PA22
 | year = 1544}}. A copy of the same table with two more entries appears on p.&nbsp;237, and another copy extended to negative powers appears on p.&nbsp;249b.</ref>

Earlier than Stifel, the 8th century [[Jain]] mathematician [[Virasena]] is credited with a precursor to the binary logarithm. Virasena's concept of ''ardhacheda'' has been defined as the number of times a given number can be divided evenly by two. This definition gives rise to a function that coincides with the binary logarithm on the powers of two,<ref>{{citation|page=[https://books.google.com/books?id=ymud91nTc9YC&pg=PA352 352]|title=The Crest of the Peacock: Non-European Roots of Mathematics|first=G. G.|last=Joseph|edition=3rd|publisher=Princeton University Press|year=2011|title-link=The Crest of the Peacock: Non-European Roots of Mathematics}}.</ref> but it is different for other integers, giving the [[p-adic order|2-adic order]] rather than the logarithm.<ref>See, e.g., {{citation|title=Cryptographic Applications of Analytic Number Theory: Complexity Lower Bounds and Pseudorandomness|volume=22|series=Progress in Computer Science and Applied Logic|first=Igor|last=Shparlinski|publisher=Birkhäuser|year=2013|isbn=978-3-0348-8037-4|page=35|url=https://books.google.com/books?id=z635BwAAQBAJ&pg=PA35}}.</ref>

The modern form of a binary logarithm, applying to any number (not just powers of two) was considered explicitly by [[Leonhard Euler]] in 1739. Euler established the application of binary logarithms to music theory, long before their applications in information theory and computer science became known. As part of his work in this area, Euler published a table of binary logarithms of the integers from 1 to 8, to seven decimal digits of accuracy.<ref>{{citation
 | last = Euler | first = Leonhard | author-link = Leonhard Euler
 | contribution = Chapter VII. De Variorum Intervallorum Receptis Appelationibus
 | language = Latin
 | pages = 102–112
 | publisher = Saint Petersburg Academy
 | title = Tentamen novae theoriae musicae ex certissismis harmoniae principiis dilucide expositae
 | url = http://eulerarchive.maa.org/pages/E033.html
 | year = 1739}}.</ref><ref>{{citation|title=London encyclopaedia; or, Universal dictionary of science, art, literature and practical mechanics: comprising a popular view of the present state of knowledge, Volume 4|first=Thomas|last=Tegg|authorlink=Thomas Tegg|year=1829|contribution=Binary logarithms|pages=142–143|url=https://books.google.com/books?id=E-ZTAAAAYAAJ&pg=PA142}}.</ref>

==Definition and properties==
The binary logarithm function may be defined as the [[inverse function]] to the [[power of two]] function, which is a strictly increasing function over the positive [[real number]]s and therefore has a unique inverse.<ref>{{citation|title=Introduction to Mathematics for Life Scientists|publisher=Springer|year=2012|first=E.|last=Batschelet|isbn=978-3-642-96080-2|url=https://books.google.com/books?id=vbT0CAAAQBAJ&pg=PA128|page=128}}.</ref>
Alternatively, it may be defined as {{math|ln ''n''/ln 2}}, where {{math|ln}} is the [[natural logarithm]], defined in any of its standard ways. Using the [[complex logarithm]] in this definition allows the binary logarithm to be extended to the [[complex number]]s.<ref>For instance, [[Microsoft Excel]] provides the <code>IMLOG2</code> function for complex binary logarithms: see {{citation|title=Excel Scientific and Engineering Cookbook|first=David M.|last=Bourg|publisher=O'Reilly Media|year=2006|isbn=978-0-596-55317-3|page=232|url=https://books.google.com/books?id=uKctiVg2dyIC&pg=PT248}}.</ref>

As with other logarithms, the binary logarithm obeys the following equations, which can be used to simplify formulas that combine binary logarithms with multiplication or exponentiation:<ref>{{citation|title=Algebra for College Students|first1=Bernard|last1=Kolman|first2=Arnold|last2=Shapiro|publisher=Academic Press|year=1982|isbn=978-1-4832-7121-7|url=https://books.google.com/books?id=i7vSBQAAQBAJ&pg=PA334|pages=334–335|contribution=11.4 Properties of Logarithms}}.</ref>
:<math>\log_2 xy=\log_2 x + \log_2 y</math>
:<math>\log_2\frac{x}{y}=\log_2 x - \log_2 y</math>
:<math>\log_2 x^y = y\log_2 x.</math>
For more, see [[list of logarithmic identities]].

==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
| origyear = 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 | authorlink=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>{{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">{{cite book |title=Einführung in die digitale Datenverarbeitung |language=German |trans-title=Introduction to digital information processing |author-first=Hans Jörg |author-last=Tafel |publisher=[[Carl Hanser Verlag]] |date=1971 |location=[[RWTH]], Aachen, Germany |publication-place=Munich, Germany |isbn=3-446-10569-7 |pages=20–21}}</ref><ref name="Tietze_Schenk_1999">{{cite book |title=Halbleiter-Schaltungstechnik |language=German |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=1370}}</ref><ref name="Bauer_2009">{{cite book
 | 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>

==Applications==

===Information theory===
The number of digits ([[bit]]s) in the [[binary representation]] of a positive integer {{mvar|n}} is the [[Floor and ceiling functions|integral part]] of {{math|1 + log<sub>2</sub>{{hsp}}''n''}}, i.e.<ref name="sw11"/>

:<math> \lfloor \log_2 n\rfloor + 1. </math>

In information theory, the definition of the amount of [[self-information]] and [[information entropy]] is often expressed with the binary logarithm, corresponding to making the bit the fundamental unit of information. However, the [[natural logarithm]] and the [[Nat (unit)|nat]] are also used in alternative notations for these definitions.<ref>{{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=https://books.google.com/books?id=tBuI_6MQTcwC&pg=PA3}}.</ref>

===Combinatorics===
[[File:SixteenPlayerSingleEliminationTournamentBracket.svg|thumb|upright=1.2|A 16-player [[single elimination]] [[Bracket (tournament)|tournament bracket]] with the structure of a [[complete binary tree]]. The height of the tree (number of rounds of the tournament) is the binary logarithm of the number of players, rounded up to an integer.]]
Although the natural logarithm is more important than the binary logarithm in many areas of pure mathematics such as [[number theory]] and [[mathematical analysis]],<ref>{{citation|title=Taming the Infinite|first=Ian|last=Stewart|authorlink=Ian Stewart (mathematician)|publisher=Quercus|year=2015|isbn=9781623654733|quote=in advanced mathematics and science the only logarithm of importance is the natural logarithm|page=120|url=https://books.google.com/books?id=u4HPBAAAQBAJ&pg=PT120}}.</ref> the binary logarithm has several applications in [[combinatorics]]:
*Every [[binary tree]] with {{mvar|n}} leaves has height at least {{math|log<sub>2</sub>{{hsp}}''n''}}, with equality when {{mvar|n}} is a [[power of two]] and the tree is a [[complete binary tree]].<ref>{{citation
 | last = Leiss | first = Ernst L.
 | isbn = 978-1-4200-1170-8
 | page = 28
 | publisher = CRC Press
 | title = A Programmer's Companion to Algorithm Analysis
 | url = https://books.google.com/books?id=E6BNGFQ6m_IC&pg=RA2-PA28
 | year = 2006}}.</ref> Relatedly, the [[Strahler number]] of a river system with {{mvar|n}} tributary streams is at most {{math|log<sub>2</sub>{{hsp}}''n'' + 1}}.<ref>{{citation
 | last1 = Devroye | first1 = L. | author1-link = Luc Devroye
 | last2 = Kruszewski | first2 = P.
 | issue = 5
 | journal = RAIRO Informatique Théorique et Applications
 | mr = 1435732
 | pages = 443–456
 | title = On the Horton–Strahler number for random tries
 | url = https://eudml.org/doc/92635
 | volume = 30
 | year = 1996| doi = 10.1051/ita/1996300504431 }}.</ref>
*Every [[family of sets]] with {{mvar|n}} different sets has at least {{math|log<sub>2</sub>{{hsp}}''n''}} elements in its union, with equality when the family is a [[power set]].<ref>Equivalently, a family with {{mvar|k}} distinct elements has at most {{math|2<sup>''k''</sup>}} distinct sets, with equality when it is a power set.</ref>
*Every [[partial cube]] with {{mvar|n}} vertices has isometric dimension at least {{math|log<sub>2</sub>{{hsp}}''n''}}, and has at most  {{math|{{sfrac|1|2}} ''n'' log<sub>2</sub>{{hsp}}''n''}} edges, with equality when the partial cube is a [[hypercube graph]].<ref>{{citation
 | last = Eppstein | first = David | authorlink = David Eppstein
 | arxiv = cs.DS/0402028
 | doi = 10.1016/j.ejc.2004.05.001
 | issue = 5
 | journal = European Journal of Combinatorics
 | mr = 2127682
 | pages = 585–592
 | title = The lattice dimension of a graph
 | volume = 26
 | year = 2005}}.</ref>
*According to [[Ramsey's theorem]], every {{mvar|n}}-vertex [[undirected graph]] has either a [[Clique (graph theory)|clique]] or an [[Independent set (graph theory)|independent set]] of size logarithmic in {{mvar|n}}. The precise size that can be guaranteed is not known, but the best bounds known on its size involve binary logarithms. In particular, all graphs have a clique or independent set of size at least {{math|{{sfrac|1|2}} log<sub>2</sub>{{hsp}}''n'' (1 − ''o''(1))}} and almost all graphs do not have a clique or independent set of size larger than {{math|2 log<sub>2</sub>{{hsp}}''n'' (1 + ''o''(1))}}.<ref>{{citation
 | last1 = Graham | first1 = Ronald L. | author1-link = Ronald Graham
 | last2 = Rothschild | first2 = Bruce L. | author2-link = Bruce Lee Rothschild
 | last3 = Spencer | first3 = Joel H. | author3-link = Joel Spencer
 | page = 78
 | publisher = Wiley-Interscience
 | title = Ramsey Theory
 | year = 1980}}.</ref>
*From a mathematical analysis of the [[Gilbert–Shannon–Reeds model]] of random shuffles, one can show that the number of times one needs to shuffle an {{mvar|n}}-card deck of cards, using [[riffle shuffle]]s, to get a distribution on permutations that is close to uniformly random, is approximately {{math|{{sfrac|3|2}} log<sub>2</sub>{{hsp}}''n''}}. This calculation forms the basis for a recommendation that 52-card decks should be shuffled seven times.<ref>{{citation
 | last1 = Bayer | first1 = Dave | author1-link = Dave Bayer
 | last2 = Diaconis | first2 = Persi | author2-link = Persi Diaconis
 | issue = 2
 | journal = The Annals of Applied Probability
 | jstor = 2959752
 | mr = 1161056
 | pages = 294–313
 | title = Trailing the dovetail shuffle to its lair
 | volume = 2
 | year = 1992 | doi=10.1214/aoap/1177005705}}.</ref>

===Computational complexity===
[[File:Binary search into array - example.svg|thumb|upright=1.2|[[Binary search]] in a sorted array, an algorithm whose time complexity involves binary logarithms]]
The binary logarithm also frequently appears in the [[analysis of algorithms]],<ref name="gt02"/> not only because of the frequent use of binary number arithmetic in algorithms, but also because binary logarithms occur in the analysis of algorithms based on two-way branching.<ref name="knuth"/>  If a problem initially has {{mvar|n}} choices for its solution, and each iteration of the algorithm reduces the number of choices by a factor of two, then the number of iterations needed to select a single choice is again the integral part of {{math|log<sub>2</sub>{{hsp}}''n''}}. This idea is used in the analysis of several [[algorithm]]s and [[data structure]]s. For example, in [[binary search]], the size of the problem to be solved is halved with each iteration, and therefore roughly {{math|log<sub>2</sub>{{hsp}}''n''}} iterations are needed to obtain a solution for a problem of size {{math|n}}.<ref>{{citation |title=Algorithms and Data Structures: The Basic Toolbox|first1=Kurt|last1=Mehlhorn|author1-link=Kurt Mehlhorn|first2=Peter|last2=Sanders|author2-link=Peter Sanders (computer scientist)|publisher=Springer|year=2008|isbn=978-3-540-77977-3|pages=34–36|contribution=2.5 An example – binary search|url=http://people.mpi-inf.mpg.de/~mehlhorn/ftp/Toolbox/Introduction.pdf}}.</ref> Similarly, a perfectly balanced [[binary search tree]] containing {{mvar|n}} elements has height {{math|log<sub>2</sub>(''n'' + 1) − 1}}.<ref>{{citation|title=Applied Combinatorics|first1=Fred|last1=Roberts|author1-link=Fred S. Roberts|first2=Barry|last2=Tesman|edition=2nd|publisher=CRC Press|year=2009|isbn=978-1-4200-9983-6|page=206|url=https://books.google.com/books?id=szBLJUhmYOQC&pg=PA206}}.</ref>

The running time of an algorithm is usually expressed in [[big O notation]], which is used to simplify expressions by omitting their constant factors and lower-order terms. Because logarithms in different bases differ from each other only by a constant factor, algorithms that run in {{math|''O''(log<sub>2</sub>{{hsp}}''n'')}} time can also be said to run in, say, {{math|''O''(log<sub>13</sub> ''n'')}} time. The base of the logarithm in expressions such as {{math|''O''(log ''n'')}} or {{math|''O''(''n'' log ''n'')}} is therefore not important and can be omitted.<ref name="clrs"/><ref>{{citation|title=Introduction to the Theory of Computation|first=Michael|last=Sipser|authorlink=Michael Sipser|edition=3rd|publisher=Cengage Learning|year=2012|isbn=9781133187790|contribution=Example 7.4|url=https://books.google.com/books?id=P3f6CAAAQBAJ&pg=PA277|pages=277–278}}.</ref> However, for logarithms that appear in the exponent of a time bound, the base of the logarithm cannot be omitted. For example, {{math|''O''(2<sup>log<sub>2</sub>{{hsp}}''n''</sup>)}} is not the same as {{math|''O''(2<sup>ln ''n''</sup>)}} because the former is equal to {{math|''O''(''n'')}} and the latter to {{math|''O''(''n''<sup>0.6931...</sup>)}}.

Algorithms with running time {{math|''O''(''n''&nbsp;log&nbsp;''n'')}} are sometimes called [[linearithmic]].<ref>{{harvtxt|Sedgewick|Wayne|2011}}, [https://books.google.com/books?id=MTpsAQAAQBAJ&pg=PA186 p.&nbsp;186].</ref> Some examples of algorithms with running time {{math|''O''(log ''n'')}} or {{math|''O''(''n'' log ''n'')}} are:

*[[quicksort|Average time quicksort]] and other [[comparison sort]] algorithms<ref>Cormen et al., p.&nbsp;156; Goodrich & Tamassia, p.&nbsp;238.</ref>
*Searching in balanced [[binary search tree]]s<ref>Cormen et al., p.&nbsp;276; Goodrich & Tamassia, p.&nbsp;159.</ref>
*[[Exponentiation by squaring]]<ref>Cormen et al., pp.&nbsp;879–880; Goodrich & Tamassia, p.&nbsp;464.</ref>
*[[Longest increasing subsequence]]<ref>{{citation|title=How to Think About Algorithms|first=Jeff|last=Edmonds|publisher=Cambridge University Press|year=2008|isbn=978-1-139-47175-6|page=302|url=https://books.google.com/books?id=hGuixQMQS_0C&pg=PT280}}.</ref>
Binary logarithms also occur in the exponents of the time bounds for some [[divide and conquer algorithm]]s, such as the [[Karatsuba algorithm]] for multiplying {{mvar|n}}-bit numbers in time {{math|''O''(''n''<sup>log<sub>2</sub>{{hsp}}3</sup>)}},<ref>Cormen et al., p.&nbsp;844; Goodrich & Tamassia, p.&nbsp;279.</ref>
and the [[Strassen algorithm]] for multiplying {{math|''n'' &times; ''n''}} matrices in time
{{math|''O''(''n''<sup>log<sub>2</sub>{{hsp}}7</sup>)}}.<ref>Cormen et al., section 28.2.</ref> The occurrence of binary logarithms in these running times can be explained by reference to the [[master theorem (analysis of algorithms)|master theorem for divide-and-conquer recurrences]].

===Bioinformatics===
[[File:Mouse cdna microarray.jpg|thumb|upright=1|A [[microarray]] for approximately 8700 genes. The expression rates of these genes are compared using binary logarithms.]]
In [[bioinformatics]], [[microarray]]s are used to measure how strongly different genes are expressed in a sample of biological material. Different rates of expression of a gene are often compared by using the binary logarithm of the ratio of expression rates: the log ratio of two expression rates is defined as the binary logarithm of the ratio of the two rates. Binary logarithms allow for a convenient comparison of expression rates: a doubled expression rate can be described by a log ratio of {{math|1}}, a halved expression rate can be described by a log ratio of {{math|−1}}, and an unchanged expression rate can be described by a log ratio of zero, for instance.<ref>{{citation|title=Microarray Gene Expression Data Analysis: A Beginner's Guide|first1=Helen|last1=Causton|first2=John|last2=Quackenbush|first3=Alvis|last3=Brazma|publisher=John Wiley & Sons|year=2009|isbn=978-1-4443-1156-3|pages=49–50|url=https://books.google.com/books?id=bg6D_7mdG70C&pg=PA49}}.</ref>

Data points obtained in this way are often visualized as a [[scatterplot]] in which one or both of the coordinate axes are binary logarithms of intensity ratios, or in visualizations such as the [[MA plot]] and [[RA plot]] that rotate and scale these log ratio scatterplots.<ref>{{citation|title=Computational and Statistical Methods for Protein Quantification by Mass Spectrometry|first1=Ingvar|last1=Eidhammer|first2=Harald|last2=Barsnes|first3=Geir Egil|last3=Eide|first4=Lennart|last4=Martens|publisher=John Wiley & Sons|year=2012|isbn=978-1-118-49378-6|page=105|url=https://books.google.com/books?id=3Z3VbhLz6pMC&pg=PA105}}.</ref>

===Music theory===
In [[music theory]], the [[Interval (music)|interval]] or perceptual difference between two tones is determined by the ratio of their frequencies. Intervals coming from [[rational number]] ratios with small numerators and denominators are perceived as particularly euphonious. The simplest and most important of these intervals is the [[octave]], a frequency ratio of {{math|2:1}}. The number of octaves by which two tones differ is the binary logarithm of their frequency ratio.<ref name="mga">{{citation|title=The Musician's Guide to Acoustics|first1=Murray|last1=Campbell|first2=Clive|last2=Greated|publisher=Oxford University Press|year=1994|isbn=978-0-19-159167-9|page=78|url=https://books.google.com/books?id=iiCZwwFG0x0C&pg=PA78}}.</ref>

To study [[tuning system]]s and other aspects of music theory that require finer distinctions between tones, it is helpful to have a measure of the size of an interval that is finer than an octave and is additive (as logarithms are) rather than multiplicative (as frequency ratios are). That is, if tones {{mvar|x}}, {{mvar|y}}, and {{mvar|z}} form a rising sequence of tones, then the measure of the interval from {{mvar|x}} to {{mvar|y}} plus the measure of the interval from {{mvar|y}} to {{mvar|z}} should equal the measure of the interval from {{mvar|x}} to {{mvar|z}}. Such a measure is given by the [[Cent (music)|cent]], which divides the octave into {{math|1200}} equal intervals ({{math|12}} [[semitone]]s of {{math|100}} cents each). Mathematically, given tones with frequencies {{math|''f''<sub>1</sub>}} and {{math|''f''<sub>2</sub>}}, the number of cents in the interval from {{math|''f''<sub>1</sub>}} to {{math|''f''<sub>2</sub>}} is<ref name="mga"/>
:<math>\left|1200\log_2\frac{f_1}{f_2}\right|.</math>
The [[millioctave]] is defined in the same way, but with a multiplier of {{math|1000}} instead of {{math|1200}}.<ref>{{citation|title=The Harvard Dictionary of Music|edition=4th|editor-first=Don Michael|editor-last=Randel|editor-link=Don Michael Randel|publisher=The Belknap Press of Harvard University Press|year=2003|isbn=978-0-674-01163-2|page=416|url=https://books.google.com/books?id=02rFSecPhEsC&pg=PA416}}.</ref>

===Sports scheduling===
In competitive games and sports involving two players or teams in each game or match, the binary logarithm indicates the number of rounds necessary in a [[single-elimination tournament]] required to determine a winner. For example, a tournament of {{math|4}} players requires {{math|1=log<sub>2</sub>{{hsp}}4 = 2}} rounds to determine the winner, a tournament of {{math|32}} teams requires {{math|1=log<sub>2</sub>{{hsp}}32 = 5}} rounds, etc. In this case, for {{mvar|n}} players/teams where {{mvar|n}} is not a power of 2, {{math|log<sub>2</sub>{{hsp}}''n''}} is rounded up since it is necessary to have at least one round in which not all remaining competitors play. For example, {{math|log<sub>2</sub>{{hsp}}6}} is approximately {{math|2.585}}, which rounds up to {{math|3}}, indicating that a tournament of {{math|6}} teams requires {{math|3}} rounds (either two teams sit out the first round, or one team sits out the second round). The same number of rounds is also necessary to determine a clear winner in a [[Swiss-system tournament]].<ref>{{citation|title=Introduction to Physical Education and Sport Science|first=Robert|last=France|publisher=Cengage Learning|year=2008|isbn=978-1-4180-5529-5|page=282|url=https://books.google.com/books?id=dH2nB1CX2SMC&pg=PA282}}.</ref>

===Photography===
In [[photography]], [[exposure value]]s are measured in terms of the binary logarithm of the amount of light reaching the film or sensor, in accordance with the [[Weber–Fechner law]] describing a logarithmic response of the human visual system to light. A single stop of exposure is one unit on a base-{{math|2}} logarithmic scale.<ref>{{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=https://books.google.com/books?id=IfWivY3mIgAC&pg=PA228}}.</ref><ref name="btzs">{{citation|title=Beyond the Zone System|first=Phil|last=Davis|publisher=CRC Press|year=1998|isbn=978-1-136-09294-7|page=17|url=https://books.google.com/books?id=YaVEAQAAQBAJ&pg=PA17}}.</ref> More precisely, the exposure value of a photograph is defined as
:<math>\log_2 \frac{N^2}{t}</math>
where {{mvar|N}} is the [[f-number]] measuring the [[aperture]] of the lens during the exposure, and {{mvar|t}} is the number of seconds of exposure.<ref>{{harvtxt|Allen|Triantaphillidou|2011}},  [https://books.google.com/books?id=IfWivY3mIgAC&pg=PA235 p.&nbsp;235].</ref>

Binary logarithms (expressed as stops) are also used in [[densitometry]], to express the [[dynamic range]] of light-sensitive materials or digital sensors.<ref>{{citation|title=Visual Effects Society Handbook: Workflow and Techniques|first1=Susan|last1=Zwerman|first2=Jeffrey A.|last2=Okun|publisher=CRC Press|year=2012|isbn=978-1-136-13614-6|page=205|url=https://books.google.com/books?id=3rLpAwAAQBAJ&pg=PA205}}.</ref>

==Calculation==
[[File:SR-50 early TI calculator.agr.jpg|thumb|upright=0.75|[[TI SR-50]] [[scientific calculator]] (1974). The ln and log keys are in the second row; there is no log<sub>2</sub>{{hsp}}key.]]

===Conversion from other bases===
An easy way to calculate {{math|log<sub>2</sub>{{hsp}}''n''}} on [[calculator]]s that do not have a {{math|log<sub>2</sub>}} function is to use the [[natural logarithm]] ({{math|ln}}) or the [[common logarithm]] ({{math|log}} or {{math|log<sub>10</sub>}}) functions, which are found on most [[scientific calculator]]s. The specific [[Logarithm#Change of base|change of logarithm base]] [[formulae]] for this are:<ref name="btzs"/><ref>{{citation|title=Secret History: The Story of Cryptology|first=Craig P.|last=Bauer|publisher=CRC Press|year=2013|isbn=978-1-4665-6186-1|page=332|url=https://books.google.com/books?id=EBkEGAOlCDsC&pg=PA332}}.</ref>

:<math>\log_2 n = \frac{\ln n}{\ln 2} = \frac{\log_{10} n}{\log_{10} 2},</math>
or approximately
:<math>\log_2 n \approx 1.442695\ln n \approx 3.321928\log_{10} n.</math>

===Integer rounding===
The binary logarithm can be made into a function from integers and to integers by [[rounding]] it up or down.  These two forms of integer binary logarithm are related by this formula:

:<math> \lfloor \log_2(n) \rfloor = \lceil \log_2(n + 1) \rceil - 1, \text{ if }n \ge 1.</math><ref name="Warren_2002">{{citation |title=Hacker's Delight |title-link=Hacker's Delight |first=Henry S. |last=Warren Jr. |date=2002 |edition=1st |publisher=[[Addison Wesley]] |isbn=978-0-201-91465-8 |page=215}}</ref>
The definition can be extended by defining <math> \lfloor \log_2(0) \rfloor = -1</math>. Extended in this way, this function is related to the [[number of leading zeros]] of the 32-bit unsigned binary representation of {{mvar|x}}, {{math|nlz(''x'')}}.
:<math>\lfloor \log_2(n) \rfloor = 31 - \operatorname{nlz}(n).</math><ref name="Warren_2002" />

The integer binary logarithm can be interpreted as the zero-based index of the most significant {{math|1}} bit in the input. In this sense it is the complement of the [[find first set]] operation, which finds the index of the least significant {{math|1}} bit. Many hardware platforms include support for finding the number of leading zeros, or equivalent operations, which can be used to quickly find the binary logarithm. The <code>fls</code> and <code>flsl</code> functions in the [[Linux kernel]]<ref>[https://www.kernel.org/doc/htmldocs/kernel-api/API-fls.html fls], Linux kernel API, [[kernel.org]], retrieved 2010-10-17.</ref> and in some versions of the [[libc]] software library also compute the binary logarithm (rounded up to an integer, plus one).

===Iterative approximation===

For a general [[positive number|positive real number]], the binary logarithm may be computed in two parts.<ref name="ml73">{{citation
 | last1 = Majithia | first1 = J. C.
 | last2 = Levan | first2 = D.
 | doi = 10.1109/PROC.1973.9318
 | issue = 10
 | journal = Proceedings of the IEEE
 | pages = 1519–1520
 | title = A note on base-2 logarithm computations
 | volume = 61
 | year = 1973}}.</ref>
First, one computes the [[Floor and ceiling functions|integer part]], <math>\lfloor\log_2 x\rfloor</math> (called the characteristic of the logarithm).
This reduces the problem to one where the argument of the logarithm is in a restricted range, the interval [1,&nbsp;2), simplifying the second step of computing the fractional part (the mantissa of the logarithm).
For any {{math|''x'' > 0}}, there exists a unique integer {{mvar|n}} such that {{math|2<sup>''n''</sup> ≤ ''x'' < 2<sup>''n''+1</sup>}}, or equivalently {{math|1 ≤ 2<sup>−''n''</sup>''x'' < 2}}.  Now the integer part of the logarithm is simply {{mvar|n}}, and the fractional part is {{math|log<sub>2</sub>(2<sup>−''n''</sup>''x'')}}.<ref name="ml73"/>  In other words:
:<math>\log_2 x = n + \log_2 y \quad\text{where } y = 2^{-n}x \text{ and } y \in [1,2)</math>
For normalized [[floating-point]] numbers, the integer part is given by the floating-point exponent,<ref>{{citation |title=Production Rendering: Design and Implementation |first=Ian |last=Stephenson |publisher=Springer-Verlag |year=2005 |isbn=978-1-84628-085-6 |url=https://books.google.com/books?id=BCC5aTR34C4C&pg=PA270 |pages=270–273 |contribution=9.6 Fast Power, Log2, and Exp2 Functions}}.</ref> and for integers it can be determined by performing a [[count leading zeros]] operation.<ref name="Warren_2013">{{Citation |title=Hacker's Delight |title-link=Hacker's Delight |first=Henry S. |last=Warren Jr. |date=2013 |orig-year=2002 |edition=2nd |publisher=[[Addison Wesley]] – [[Pearson Education, Inc.]] |isbn=978-0-321-84268-8 |id=0-321-84268-5 |page=291 |contribution=11-4: Integer Logarithm|contribution-url=https://books.google.com/books?id=XD9iAwAAQBAJ&pg=PA291}}.</ref>

The fractional part of the result is {{math|log<sub>2</sub>{{hsp}}''y''}} and can be computed iteratively, using only elementary multiplication and division.<ref name="ml73"/>
The algorithm for computing the fractional part can be described in [[pseudocode]] as follows:
# Start with a real number {{mvar|y}} in the half-open interval {{math|[1,&nbsp;2)}}.  If {{math|1=''y'' = 1}}, then the algorithm is done, and the fractional part is zero.
# Otherwise, square {{mvar|y}} repeatedly until the result {{mvar|z}} lies in the interval {{math|[2,&nbsp;4)}}.  Let {{mvar|m}} be the number of squarings needed.  That is, {{math|1=''z'' = ''y''<sup>2<sup>''m''</sup></sup>}} with {{mvar|m}} chosen such that {{mvar|z}} is in {{math|[2,&nbsp;4)}}.
# Taking the logarithm of both sides and doing some algebra:
#:<math>\begin{align}
 \log_2 z &= 2^m \log_2 y \\
 \log_2 y &= \frac{ \log_2 z }{ 2^m } \\
          &= \frac{ 1 + \log_2(z/2) }{ 2^m } \\
          &= 2^{-m} + 2^{-m}\log_2(z/2).
\end{align}</math>
# Once again {{math|''z''/2}} is a real number in the interval {{math|[1,&nbsp;2)}}. Return to step 1 and compute the binary logarithm of {{math|''z''/2}} using the same method.

The result of this is expressed by the following recursive formulas, in which <math>m_i</math> is the number of squarings required in the ''i''-th iteration of the algorithm:
:<math>\begin{align}
 \log_2 x &= n + 2^{-m_1} \left( 1 + 2^{-m_2} \left( 1 + 2^{-m_3} \left( 1 + \cdots \right)\right)\right) \\
          &= n + 2^{-m_1} + 2^{-m_1-m_2} + 2^{-m_1-m_2-m_3} + \cdots
\end{align}</math>

In the special case where the fractional part in step 1 is found to be zero, this is a ''finite'' sequence terminating at some point.  Otherwise, it is an [[infinite series]] that [[convergent series|converge]]s according to the [[ratio test]], since each term is strictly less than the previous one (since every {{math|''m''<sub>''i''</sub> > 0}}).  For practical use, this infinite series must be truncated to reach an approximate result.  If the series is truncated after the {{mvar|i}}-th term, then the error in the result is less than {{math|2<sup>−(''m''<sub>1</sub> + ''m''<sub>2</sub>{{hsp}}+ ... + ''m''<sub>''i''</sub>)</sup>}}.<ref name="ml73"/>

It is possible to square the number repeatedly regardless of whether it reaches interval {{math|[2,&nbsp;4)}} or beyond, and extract the bits of the result from the number anyways. For example, <math>y^2 \, 2^{-p} \in [1, 2)</math> gives solutions <math>p = 0, 1</math>. Finding <math>p = 1</math> is equivalent to finding <math>m_1 = 1</math>, and finding otherwise shows that <math>m_1 > 1</math>. Using <math>y^4 \, 2^{-p} \in [1, 2)</math> instead gives <math>p \in [0, 4)</math>, where if {{mvar|p}} is 0, <math>m_1 > 2</math>, when it is 1, <math>m_1 = 2</math>, when it is 2, <math>m_1 = 1,\ m_2 > 1</math>, and when it is 3, <math>m_1 = m_2 = 1</math>.

When looking at these values for <math>m_{1..n}</math> from <math>y^{(2^n)} \, 2^{-p} \in [1, 2)</math> in terms of the result, it can be seen that <math>p \in [0, 2^n)</math> contains the bits of the result starting from the <math>2^{-1}</math> being the highest bit of {{mvar|p}}. This works well in [[IEEE 754]] binary64 or binary32 numbers by extracting the base-2 exponent. For binary64, {{mvar|y}} can be squared 10 times at most (otherwise the result can overflow to infinity), and maximum of 7 times for binary32 (the maximum is the number of exponent bits in the format, minus one, as the exponent needs to be able to store <math>2^n</math>). After performing the multiplications, the number needs to be reduced back to range {{math|[1,&nbsp;2)}}, which is simply setting the exponent to 0. The same operation can then be performed all over again, and the new bits appended to the result. Using this format, instead of performing 1 multiplication and comparison at a time, up to 10 (or 7) multiplications and then some bit-shifting can be done at a time.

===Software library support===
The <code>log2</code> function is included in the standard [[C mathematical functions]]. The default version of this function takes [[double precision]] arguments but variants of it allow the argument to be single-precision or to be a [[long double]].<ref>{{citation | url=http://www.open-std.org/jtc1/sc22/wg14/www/docs/n1124.pdf | title=ISO/IEC 9899:1999 specification | page=226| contribution = 7.12.6.10 The log2 functions }}.</ref> In [[MATLAB]], the argument to the <code>log2</code> function is allowed to be a [[negative number]], and in this case the result will be a [[complex number]].<ref>{{citation|title=The Matlab® 5 Handbook|first1=Darren|last1=Redfern|first2=Colin|last2=Campbell|publisher=Springer-Verlag|year=1998|isbn=978-1-4612-2170-8|url=https://books.google.com/books?id=KQDaBwAAQBAJ&pg=PA141|page=141}}.</ref>

==References==
{{reflist}}

==External links==
*{{mathworld|id=BinaryLogarithm|title=Binary Logarithm}}
*{{citation|url=http://graphics.stanford.edu/~seander/bithacks.html#IntegerLog|title=Find the log base 2 of an N-bit integer in O(lg(N)) operations|work=Bit Twiddling Hacks|first=Sean Eron|last=Anderson|publisher=Stanford University|date=December 12, 2003|accessdate=2015-11-25}}
*[http://longnow.org/essays/richard-feynman-and-connection-machine/ Feynman and the Connection Machine]

[[Category:Binary arithmetic]]
[[Category:Calculus]]
[[Category:Logarithms]]