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Binary logarithm

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The binary logarithm of a positive real number is the exponent such that . Since , we have , where without the subscript is the natural logarithm ( is approximately 0.69314718..., see A002162). For example, is approximately 1.65149612947... The binary logarithm of is an integer only when is an integer power of 2 (see A000079) or the reciprocal of an integer power of 2 (in the latter case a negative integer).

Occasionally, a couple other notations are used for the binary logarithm, specifically and . However, for maximum clarity, it is best to use .