## logarithm
The logarithm of a number or variable x to base b, log_{b} x, is the exponent of b needed to give x. The bases most commonly used in mathematics are e and 10. A logarithm to base e, written as log x or ln x, is known as a natural logarithm. (For example, log_{e}10 is approximately 2.30258.) A logarithm
to base 10 is written as log_{10} x and is known as a common
logarithm. Since a^{0} = 1 for any a; log 1 = 0 for all bases.
In order to multiply two numbers together, one uses the fact that a = ^{x}.a^{y}a^{x + y}, and hence log (x.y)
= log x + log y. We therefore look up the values of log x and log y in logarithmic tables, add these values, and
then use the tables again to find the number whose logarithm is equal to
the result of the addition. Similarly, since a/^{x}a = ^{y}a = log(^{x = y}x/y); and since (a)^{x} = ^{y}a, log ^{xy}x = ^{y}y.log x.
Log_{x}x = 1 since x^{1} = x. To calculate in decimal numbers, logarithm tables need be compiled only for the numbers between 0 and 9.999 (in four-figure tables; in five-figure tables include 9.9999, and so on to as many figures as required). Larger numbers are expressed by adding a whole number (integer) called the characteristic, which represents in base-10 logs the corresponding power. The four-figure logarithm to the base 10 (written log _{10}) of 2, for example, is 0.3010. The log of 200 is 2.3010 and of 2,000 is 3.3010 (200 is 10^{2} × 10^{0.3010} = 10^{2.3010}, and 2,000 is 10^{3} × 10^{0.3010} = 10^{3.3010}). The antilogarithm of a number x is the number
whose logarithm is x; that is, if log y = x,
then y is the antilogarithm of x. A logarithmic curve is the plotting of a function of the form f(x) = log x (i.e., a logarithmic
function). Logarithms, as an aid to calculation, were introduced by John Napier in 1614 and developed by Henry Briggs ## Related entry• slide rule## Related category• FUNCTIONS | |||||||

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