Decadic logarithm

In mathematics, the common logarithm is the logarithm with base 10. It is also known as the decadic logarithm and also as the decimal logarithm, named after its base, or Briggsian logarithm, after Henry Briggs, an English mathematician who pioneered its use, as well as "standard logarithm". Historically, it was known as logarithmus decimalis or logarithmus decadis. It is indicated by log₁₀(x), or sometimes Log(x) with a capital L (however, this notation is ambiguous since it can also mean the complex natural logarithmic multi-valued function). On calculators it is usually "log", but mathematicians usually mean natural logarithm (logarithm with base e ≈ 2.71828) rather than common logarithm when they write "log". To mitigate this ambiguity the ISO 80000 specification recommends that log₁₀(x) should be written lg (x) and log_e(x) should be ln (x).

Before the early 1970s, handheld electronic calculators were not available and mechanical calculators capable of multiplication were bulky, expensive and not widely available. Instead, tables of base-10 logarithms were used in science, engineering and navigation when calculations required greater accuracy than could be achieved with a slide rule. Use of logarithms avoided laborious and error prone paper and pencil multiplications and divisions. Because logarithms were so useful, tables of base-10 logarithms were given in appendices of many text books. Mathematical and navigation handbooks included tables of the logarithms of trigonometric functions as well. See log table for the history of such tables.

An important property of base-10 logarithms which makes them so useful in calculation is that the logarithm of numbers greater than one which differ by a factor of a power of ten all have the same fractional part. The fractional part is known as the mantissa. Thus log tables need only show the fractional part. Tables of common logarithms typically listed the mantissa, to 4 or 5 decimal places or more, of each number in a range, e.g. 1000 to 9999. Such a range would cover all possible values of the mantissa.

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