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Logarithms can be used to make calculations easier. For example, two numbers can be multiplied just by using a logarithm table and adding. These are often known as logarithmic properties, which are documented in the table below. [2] The first three operations below assume that x = b c and/or y = b d, so that log b (x) = c and log b (y) = d.
The law of iterated logarithms operates "in between" the law of large numbers and the central limit theorem.There are two versions of the law of large numbers — the weak and the strong — and they both state that the sums S n, scaled by n −1, converge to zero, respectively in probability and almost surely:
In c. 1622, William Oughtred combined two handheld Gunter rules to make a calculating device that was essentially the first slide rule. [9] The logarithm function became a staple of mathematical analysis, but printed tables of logarithms gradually diminished in importance in the twentieth century as multiplying mechanical calculators and, later ...
Another critical application was the slide rule, a pair of logarithmically divided scales used for calculation. The non-sliding logarithmic scale, Gunter's rule, was invented shortly after Napier's invention. William Oughtred enhanced it to create the slide rule—a pair of logarithmic scales movable with respect to each other. Numbers are ...
The logarithmic decrement can be obtained e.g. as ln(x 1 /x 3).Logarithmic decrement, , is used to find the damping ratio of an underdamped system in the time domain.. The method of logarithmic decrement becomes less and less precise as the damping ratio increases past about 0.5; it does not apply at all for a damping ratio greater than 1.0 because the system is overdamped.
The following is a list of integrals (antiderivative functions) of logarithmic functions. For a complete list of integral functions, see list of integrals. Note: x > 0 is assumed throughout this article, and the constant of integration is omitted for simplicity.
In addition, log structures serve to define the mixed Hodge structure on any smooth complex variety X, by taking a compactification with boundary a normal crossings divisor D, and writing down the corresponding logarithmic de Rham complex. [2] Log objects also naturally appear as the objects at the boundary of moduli spaces, i.e. from ...
Analogously, in any group G, powers b k can be defined for all integers k, and the discrete logarithm log b a is an integer k such that b k = a. In arithmetic modulo an integer m , the more commonly used term is index : One can write k = ind b a (mod m ) (read "the index of a to the base b modulo m ") for b k ≡ a (mod m ) if b is a primitive ...