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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the common logarithm is the
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 ...
with base 10. It is also known as the decadic logarithm and as the decimal logarithm, named after its base, or Briggsian logarithm, after
Henry Briggs Henry Briggs may refer to: *Henry Briggs (mathematician) (1561–1630), English mathematician *Henry Perronet Briggs (1793–1844), English painter *Henry George Briggs (1824–1872), English merchant, traveller, and orientalist *Henry Shaw 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 , , or sometimes with a capital (however, this notation is ambiguous, since it can also mean the complex natural logarithmic multi-valued function). On
calculator An electronic calculator is typically a portable electronic device used to perform calculations, ranging from basic arithmetic to complex mathematics. The first solid-state electronic calculator was created in the early 1960s. Pocket-sized ...
s, it is printed as "log", but mathematicians usually mean
natural logarithm The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
(logarithm with base e ≈ 2.71828) rather than common logarithm when they write "log". To mitigate this ambiguity, the ISO 80000 specification recommends that should be written , and should be . Before the early 1970s, handheld electronic calculators were not available, and
mechanical calculator A mechanical calculator, or calculating machine, is a mechanical device used to perform the basic operations of arithmetic automatically, or (historically) a simulation such as an analog computer or a slide rule. Most mechanical calculators w ...
s 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 The slide rule is a mechanical analog computer which is used primarily for multiplication and division, and for functions such as exponents, roots, logarithms, and trigonometry. It is not typically designed for addition or subtraction, which ...
. By turning multiplication and division to addition and subtraction, 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 textbooks. Mathematical and navigation handbooks included tables of the logarithms of
trigonometric function In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
s as well. For the history of such tables, see log table.


Mantissa and characteristic

An important property of base-10 logarithms, which makes them so useful in calculations, is that the logarithm of numbers greater than 1 that differ by a factor of a power of 10 all have the same fractional part. The fractional part is known as the mantissa.This use of the word ''mantissa'' stems from an older, non-numerical, meaning: a minor addition or supplement, e.g., to a text. Nowadays, the word ''mantissa'' is generally used to describe the fractional part of a
floating-point In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can ...
number on computers, though the recommended term is significand.
Thus, log tables need only show the fractional part. Tables of common logarithms typically listed the mantissa, to four or five decimal places or more, of each number in a range, e.g. 1000 to 9999. The integer part, called the characteristic, can be computed by simply counting how many places the decimal point must be moved, so that it is just to the right of the first significant digit. For example, the logarithm of 120 is given by the following calculation: :\log_(120) = \log_\left(10^2 \times 1.2\right) = 2 + \log_(1.2) \approx 2 + 0.07918. The last number (0.07918)—the fractional part or the mantissa of the common logarithm of 120—can be found in the table shown. The location of the decimal point in 120 tells us that the integer part of the common logarithm of 120, the characteristic, is 2.


Negative logarithms

Positive numbers less than 1 have negative logarithms. For example, :\log_(0.012) = \log_\left(10^ \times 1.2\right) = -2 + \log_(1.2) \approx -2 + 0.07918 = -1.92082. To avoid the need for separate tables to convert positive and negative logarithms back to their original numbers, one can express a negative logarithm as a negative integer characteristic plus a positive mantissa. To facilitate this, a special notation, called ''bar notation,'' is used: :\log_(0.012) \approx \bar + 0.07918 = -1.92082. The bar over the characteristic indicates that it is negative, while the mantissa remains positive. When reading a number in bar notation out loud, the symbol \bar is read as "bar ", so that \bar.07918 is read as "bar 2 point 07918…". An alternative convention is to express the logarithm modulo 10, in which case :\log_(0.012) \approx 8.07918 \bmod 10, with the actual value of the result of a calculation determined by knowledge of the reasonable range of the result.For example, gives (beginning of section 8) \log b = 6.51335464, \log e = 8.9054355. From the context, it is understood that b = 10^, the minor radius of the earth ellipsoid in
toise A toise (; symbol: T) is a unit of measure for length, area and volume originating in pre-revolutionary France. In North America, it was used in colonial French establishments in early New France, French Louisiana (''Louisiane''), Acadia (''Acadi ...
(a large number), whereas e = 10^, the eccentricity of the earth ellipsoid (a small number).
The following example uses the bar notation to calculate 0.012 × 0.85 = 0.0102: :\begin \text & \log_(0.012) \approx\bar.07918\\ \text\;\;\log_(0.85) &= \log_\left(10^\times 8.5\right) = -1 + \log_(8.5) &\approx -1 + 0.92942 = \bar.92942\\ \log_(0.012 \times 0.85) &= \log_(0.012) + \log_(0.85) &\approx \bar.07918 + \bar.92942\\ &= (-2 + 0.07918) + (-1 + 0.92942) &= -(2 + 1) + (0.07918 + 0.92942)\\ &= -3 + 1.00860 &= -2 + 0.00860\;^*\\ &\approx \log_\left(10^\right) + \log_(1.02) &= \log_(0.01 \times 1.02)\\ &= \log_(0.0102). \end * This step makes the mantissa between 0 and 1, so that its
antilog In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 o ...
(10) can be looked up. The following table shows how the same mantissa can be used for a range of numbers differing by powers of ten: Note that the mantissa is common to all of the . This holds for any positive real number x because :\log_\left(x \times10^i\right) = \log_(x) + \log_\left(10^i\right) = \log_(x) + i. Since is a constant, the mantissa comes from \log_(x), which is constant for given x. This allows a table of logarithms to include only one entry for each mantissa. In the example of , 0.698 970 (004 336 018 ...) will be listed once indexed by 5 (or 0.5, or 500, etc.).


History

Common logarithms are sometimes also called "Briggsian logarithms" after
Henry Briggs Henry Briggs may refer to: *Henry Briggs (mathematician) (1561–1630), English mathematician *Henry Perronet Briggs (1793–1844), English painter *Henry George Briggs (1824–1872), English merchant, traveller, and orientalist *Henry Shaw Briggs ...
, a 17th century British mathematician. In 1616 and 1617, Briggs visited
John Napier John Napier of Merchiston (; 1 February 1550 – 4 April 1617), nicknamed Marvellous Merchiston, was a Scottish landowner known as a mathematician, physicist, and astronomer. He was the 8th Laird of Merchiston. His Latinized name was Ioan ...
at
Edinburgh Edinburgh ( ; gd, Dùn Èideann ) is the capital city of Scotland and one of its 32 council areas. Historically part of the county of Midlothian (interchangeably Edinburghshire before 1921), it is located in Lothian on the southern shore of t ...
, the inventor of what are now called natural (base-''e'') logarithms, in order to suggest a change to Napier's logarithms. During these conferences, the alteration proposed by Briggs was agreed upon; and after his return from his second visit, he published the first
chiliad 1000 or one thousand is the natural number following 999 and preceding 1001. In most English-speaking countries, it can be written with or without a comma or sometimes a period separating the thousands digit: 1,000. A group of one thousand t ...
of his logarithms. Because base-10 logarithms were most useful for computations, engineers generally simply wrote "" when they meant . Mathematicians, on the other hand, wrote "" when they meant for the natural logarithm. Today, both notations are found. Since hand-held electronic calculators are designed by engineers rather than mathematicians, it became customary that they follow engineers' notation. So the notation, according to which one writes "" when the natural logarithm is intended, may have been further popularized by the very invention that made the use of "common logarithms" far less common, electronic calculators.


Numeric value

The numerical value for logarithm to the base 10 can be calculated with the following identities: : \log_(x) = \frac \quad or \quad \log_(x) = \frac \quad or \quad \log_(x) = \frac \quad using logarithms of any available base \, B ~. as procedures exist for determining the numerical value for logarithm base (see ) and logarithm base 2 (see Algorithms for computing binary logarithms).


Derivative

The derivative of a logarithm with a base ''b'' is such that \log_b(x)=, so \log_(x)=.


See also

* Binary logarithm *
Cologarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 o ...
* Decibel * Logarithmic scale * Mantissa (floating point number) *
Napierian logarithm The term Napierian logarithm or Naperian logarithm, named after John Napier, is often used to mean the natural logarithm. Napier did not introduce this ''natural'' logarithmic function, although it is named after him. However, if it is taken to m ...


Notes


References


Bibliography

* * * {{Authority control Logarithms