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In mathematics, the binary logarithm () is the power to which the number must be raised to obtain the value . That is: : For example, the binary logarithm of is , the binary logarithm of is , the binary logarithm of is , the binary logarithm of is , the binary logarithm of is and the binary logarithm of is . The binary logarithm is the logarithm to the . The function that maps a number to its binary logarithm is the inverse function of the power of two function. As well as , alternative notations for the binary logarithm include , , , and (with a prior notation that the default base is 2) . 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 octaves 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 bits 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 tournaments, 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 using a recursive algorithm. ==History== (詳細はpowers of two have been known since antiquity; for instance they appear in 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 numbers). 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 ''Arthmetica Integra'' contains several tables that show the integers with their corresponding powers of two. Reversing the rows of these tables allow them to be interpreted as tables of binary logarithms.〔 .〕〔. A copy of the same table with two more entries appears on p. 237, and another copy extended to negative powers appears on p. 249b.〕 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,〔.〕 but it is different for other integers, giving the 2-adic order rather than the logarithm.〔See, e.g., .〕 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 more significant 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.〔.〕〔.〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「binary logarithm」の詳細全文を読む スポンサード リンク
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