What is "logarithm"

Logarithm \Log"a*rithm\ (l[o^]g"[.a]*r[i^][th]'m), n. [Gr. lo`gos word, account, proportion + 'ariqmo`s number: cf. F. logarithme.] (Math.) One of a class of auxiliary numbers, devised by John Napier, of Merchiston, Scotland (1550-1617), to abridge arithmetical calculations, by the use of addition and subtraction in place of multiplication and division.

Note: The relation of logarithms to common numbers is that of numbers in an arithmetical series to corresponding numbers in a geometrical series, so that sums and differences of the former indicate respectively products and quotients of the latter; thus, 0 1 2 3 4 Indices or logarithms 1 10 100 1000 10,000 Numbers in geometrical progression Hence, the logarithm of any given number is the exponent of a power to which another given invariable number, called the base, must be raised in order to produce that given number. Thus, let 10 be the base, then 2 is the logarithm of 100, because 10^ 2 = 100, and 3 is the logarithm of 1,000, because 10^ 3 = 1,000.

Arithmetical complement of a logarithm, the difference between a logarithm and the number ten.

Binary logarithms. See under Binary.

Common logarithms, or Brigg's logarithms, logarithms of which the base is 10; -- so called from Henry Briggs, who invented them.

Gauss's logarithms, tables of logarithms constructed for facilitating the operation of finding the logarithm of the sum of difference of two quantities from the logarithms of the quantities, one entry of those tables and two additions or subtractions answering the purpose of three entries of the common tables and one addition or subtraction. They were suggested by the celebrated German mathematician Karl Friedrich Gauss (died in 1855), and are of great service in many astronomical computations.

Hyperbolic logarithm or Napierian logarithm or Natural logarithm, a logarithm (devised by John Speidell, 1619) of which the base is e (2.718281828459045...); -- so called from Napier, the inventor of logarithms.

Logistic logarithms or Proportional logarithms, See under Logistic.

1610s, Modern Latin logarithmus, coined by Scottish mathematician John Napier (1550-1617), literally "ratio-number," from Greek logos "proportion, ratio, word" (see logos) + arithmos "number" (see arithmetic).

n. (context mathematics English) For a number $x$, the power to which a given ''base'' number must be raised in order to obtain $x$. Written $log\_b\; x$. For example, $log\_\{10\}\; 1000\; =\; 3$ because $10^3\; =\; 1000$ and $log\_2\; 16\; =\; 4$ because $2^4\; =\; 16$.

n. the exponent required to produce a given number [syn: log]

In mathematics, the logarithm is the inverse operation to exponentiation. That means the logarithm of a number is the exponent to which another fixed number, the base, must be raised to produce that number. In simple cases the logarithm counts repeated multiplication. For example, the base logarithm of is , as to the power is ; the multiplication is repeated three times. More generally, exponentiation allows any positive real number to be raised to any real power, always producing a positive result, so the logarithm can be calculated for any two positive real numbers and where is not equal to . The logarithm of to base , denoted , is the unique real number such that

.

For example, as , then:

The logarithm to base (that is ) is called the common logarithm and has many applications in science and engineering. The natural logarithm has the number as its base; its use is widespread in mathematics and physics, because of its simpler derivative. The binary logarithm uses base (that is ) and is commonly used in computer science.

Logarithms were introduced by John Napier in the early 17th century as a means to simplify calculations. They were rapidly adopted by navigators, scientists, engineers, and others to perform computations more easily, using slide rules and logarithm tables. Tedious multi-digit multiplication steps can be replaced by table look-ups and simpler addition because of the fact — important in its own right — that the logarithm of a product is the sum of the logarithms of the factors:

log(xy) = log(x) + log(y),  
provided that , and are all positive and . The present-day notion of logarithms comes from Leonhard Euler, who connected them to the exponential function in the 18th century.

Logarithmic scales reduce wide-ranging quantities to tiny scopes. For example, the decibel is a unit quantifying signal power log-ratios and amplitude log-ratios (of which sound pressure is a common example). In chemistry, pH is a logarithmic measure for the acidity of an aqueous solution. Logarithms are commonplace in scientific formulae, and in measurements of the complexity of algorithms and of geometric objects called fractals. They describe musical intervals, appear in formulas counting prime numbers, inform some models in psychophysics, and can aid in forensic accounting.

In the same way as the logarithm reverses exponentiation, the complex logarithm is the inverse function of the exponential function applied to complex numbers. The discrete logarithm is another variant; it has uses in public-key cryptography.