In mathematics, the common logarithm is the logarithm with base 10. (That is why it is also known as decadic logarithm; deca means "ten".) 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 rather than common logarithm when they write "log". water flow File links The following pages link to this file: Common logarithm ... water flow File links The following pages link to this file: Common logarithm ... Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... Logarithms to various bases: is to base e, is to base 10, and is to base 1. ... The natural logarithm, invented by John Napier, is the logarithm to the base e, where e is equal to 2. ...

Before the early 1970s, hand-held electronic calculators were not yet in widespread use. Because of their utility in saving work in laborious calculations by hand on paper, tables of base-10 logarithms were found in appendices of many books. Such a table of "common logarithms" giving the logarithm of each number in the left-hand column, which ran from 1 to 10 by small increments, perhaps 0.01 or 0.001. There was no need to include numbers not between 1 and 10, since if one wanted the logarithm of, for example, 120, one would know that The 1970s decade refers to the years from 1970 to 1979, inclusive. ... A modern basic arithmetic calculator A calculator is a device for performing numerical calculations. ... Before calculators were cheap and plentiful, people would use mathematical tables —lists of numbers showing the results of calculation with varying variables— to simplify and drastically speed up computation. ... Logarithms to various bases: is to base e, is to base 10, and is to base 1. ...

The very last number ( 0.079181) -- the fractional part of the logarithm of 120, known as the mantissa of the common logarithm of 120 -- was found in the table. (This stems from an older, non-numerical, meaning of the word mantissa: a minor addition or supplement, e.g. to a text. For a more modern use of the word mantissa, see significand.) The location of the decimal point in 120 tells us that the integer part of the common logarithm of 120, called the characteristic of the common logarithm of 120, is 2. The significand (also coefficient or, more informally, mantissa) is the part of a floating-point number that contains its significant digits. ...

Similarly, for numbers less than 1 we have

The bar over the characteristic indicates that it is negative whilst the mantissa remains positive. Negative logarithm values were rarely converted to a normal negative number (−0.920819 in the example).

In addition, slide rules worked by using a logarithmic scale. The slide rule, or slipstick, is an analog computer, usually consisting of three interlocking calibrated strips and a sliding window, called the cursor. ... A logarithmic scale is a scale of measurement that uses the logarithm of a physical quantity instead of the quantity itself. ...

Common logarithms are sometimes also called Briggsian logarithms after Henry Briggs, a 17th-century British mathematician. Henry Briggs (February 1556 - January 26, 1630) was an English mathematician. ... (16th century - 17th century - 18th century - more centuries) As a means of recording the passage of time, the 17th century was that century which lasted from 1601-1700. ...

Because base-10 logarithms were called "common", and engineers often had occasion to use them, engineers often wrote "log(x)" when they meant log₁₀(x). Mathematicians, on the other hand, wrote "log(x)" when they mean log_e(x) (see natural logarithm). Today, both notations are found among mathematicians. Since hand-held electronic calculators are designed by engineers rather than mathematicians, it became customary that they follow engineers' notation. So ironically, that notation, according to which one writes "ln(x)" when the natural logarithm is intended, may have been further popularized by the very invention that made the use of "common logarithms" obsolete: electronic calculators. The natural logarithm, invented by John Napier, is the logarithm to the base e, where e is equal to 2. ...

Early electronic calculators did not have the ability to calculate logarithms, but many could extract square roots. There is a curious approximation to the common logarithm that can be made on such a calculator. Take the square root 11 times. Subtract 1. Multiply by 889. For a wide range of numbers from 10⁻¹⁷ to 10⁺¹⁸, this is accurate to within 1%. In other words:

It is based on the fact that 889 ln 10 ≈ 2048 and ln x ≈ x − 1 for x ≈ 1.

See also

• Jurij Vega
• slide rule