Binary logarithm

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the binary logarithm (log2 n) is the <a href="/facts/Exponentiation/pac6QY2g">power</a> to which the number 2 must be <a href="/facts/Exponentiation/pac6QY2g">raised</a> to obtain the value n. That is, for any real number x,

x
        =
        
          log
          
            2
          
        
        ⁡
        n
        
        ⟺
        
        
          2
          
            x
          
        
        =
        n
        .
      
    
    {\displaystyle x=\log _{2}n\quad \Longleftrightarrow \quad 2^{x}=n.}

For example, the binary logarithm of 1 is 0, the binary logarithm of 2 is 1, the binary logarithm of 4 is 2, and the binary logarithm of 32 is 5.
The binary logarithm is the <a href="/facts/Logarithm/QeXaV97q">logarithm</a> to the base 2 and is the <a href="/facts/Inverse_function/Tg5nnXlu">inverse function</a> of the <a href="/facts/Power_of_two/MA6WEpTt">power of two</a> function. There are several alternatives to the log2 notation for the binary logarithm; see the Notation section below.
Historically, the first application of binary logarithms was in <a href="/facts/Music_theory/BoekvflP">music theory</a>, by <a href="/facts/Leonhard_Euler/z2fsbjgj">Leonhard Euler</a>: the binary logarithm of a frequency ratio of two musical tones gives the number of <a href="/facts/Octave/RA4gWvkV">octaves</a> by which the tones differ. Binary logarithms can be used to calculate the length of the representation of a number in the <a href="/facts/Binary_numeral_system/a6JDSRPs">binary numeral system</a>, or the number of <a href="/facts/Bit/S972NSaD">bits</a> needed to encode a message in <a href="/facts/Information_theory/qNLH3U5P">information theory</a>. In <a href="/facts/Computer_science/lN3pEyPz">computer science</a>, they count the number of steps needed for <a href="/facts/Binary_search/FFmQzunc">binary search</a> and related algorithms. Other areas
in which the binary logarithm is frequently used include <a href="/facts/Combinatorics/k14xUoKT">combinatorics</a>, <a href="/facts/Bioinformatics/D5x2L8ee">bioinformatics</a>, the design of sports <a href="/facts/Tournament/htGM1TzT">tournaments</a>, and <a href="/facts/Photography/ezLPT43M">photography</a>.
Binary logarithms are included in the standard <a href="/facts/C_mathematical_functions/2L11HPLP">C mathematical functions</a> and other mathematical software packages.

Binary logarithm open-in-new

Binary logarithm