Editing Binary logarithm (section)

{{Short description|Exponent of a power of two}}
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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}} and is the [[inverse function]] of the [[power of two]] function. There are several alternatives to the {{math|log<sub>2</sub>}} notation for the binary logarithm; see the [[#Notation|Notation]] section below.

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.