Editing Binary logarithm (section)

==History==
{{Main|History of logarithms}}
[[File:Leonhard Euler - Jakob Emanuel Handmann (Kunstmuseum Basel).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 = la
 | 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 = la
 | 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|author-link=Thomas Tegg|year=1829|contribution=Binary logarithms|pages=142–143|url=https://books.google.com/books?id=E-ZTAAAAYAAJ&pg=PA142}}.</ref>