# Common logarithm

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In mathematics, the common logarithm is the logarithm with base 10. (That is why it is also known as decadic logarithm; deca means "ten")

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. In addition, slide rules worked by using a logarithmic scale.

Base-10 logarithms were called common logarithms. Such a table of "common logarithms" gave 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

[itex]\log_{10}120=\log_{10}(10^2\times 1.2)=2+\log_{10}1.2\cong2+0.079181.[itex]

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.

Similarly, for numbers less than 1 we have

[itex]\log_{10}0.12=\log_{10}(10^{-1}\times 1.2)=-1+\log_{10}1.2\cong-1+0.079181=\bar{1}.079181.[itex]

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).

Common logarithms are sometimes also called Briggsian logarithms after Henry Briggs, a 17th-century British mathematician.

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 loge(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.

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:

[itex]889\times(x^{1/2048} - 1) \approx \log_{10}x.[itex]

It is based on the fact that 889 ln 10 ≈ 2048

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