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Particular bases

Among all choices for the base, three are particularly common. These are b = 10, b = e (the irrational mathematical constant e ≈ 2.71828183), and b = 2 (the binary logarithm). In mathematical analysis, the logarithm base e is widespread because of analytical properties explained below. On the other hand, base 10 logarithms (the common logarithm) are easy to use for manual calculations in the decimal number system:

log 10 (10 x) = log 10 10 + log 10 x = 1 + log 10 x. {\displaystyle \log _{10}\,(\,10\,x\,)\ =\;\log _{10}10\ +\;\log _{10}x\ =\ 1\,+\,\log _{10}x\,.}

Thus, log (x) is related to the number of decimal digits of a positive integer x: The number of digits is the smallest integer strictly bigger than log (x). For example, log (5986) is approximately 3.78. The next integer above it is 4, which is the number of digits of 5986. Both the natural logarithm and the binary logarithm are used in information theory, corresponding to the use of nats or bits as the fundamental units of information, respectively. Binary logarithms are also used in computer science, where the binary system is ubiquitous; in music theory, where a pitch ratio of two (the octave) is ubiquitous and the number of cents between any two pitches is a scaled version of the binary logarithm, or log 2 times 1200, of the pitch ratio (that is, 100 cents per semitone in conventional equal temperament), or equivalently the log base 2 ; and in photography rescaled base 2 logarithms are used to measure exposure values, light levels, exposure times, lens apertures, and film speeds in "stops".

The abbreviation log x is often used when the intended base can be inferred based on the context or discipline, or when the base is indeterminate or immaterial. Common logarithms (base 10), historically used in logarithm tables and slide rules, are a basic tool for measurement and computation in many areas of science and engineering; in these contexts log x still often means the base ten logarithm. In mathematics log x usually means to the natural logarithm (base e). In computer science and information theory, log often refers to binary logarithms (base 2). The following table lists common notations for logarithms to these bases. The "ISO notation" column lists designations suggested by the International Organization for Standardization.