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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.
Exponentiation occurs in many areas of mathematics and its inverse function is often referred to as the logarithm. For example, the logarithm of a matrix is the (multi-valued) inverse function of the matrix exponential. [97] Another example is the p-adic logarithm, the inverse function of the p-adic exponential.
In mathematics, the common logarithm (aka "standard logarithm") is the logarithm with base 10. [1] It is also known as the decadic logarithm , the decimal logarithm and the Briggsian logarithm . The name "Briggsian logarithm" is in honor of the British mathematician Henry Briggs who conceived of and developed the values for the "common logarithm".
For example, one can multiply a sine that is less than 0.5 by some power of two or ten to bring it into the range [0.5,1]. After finding that logarithm in the radical table, one adds the logarithm of the power of two or ten that was used (he gives a short table), to get the required logarithm. [1]: p. 36
[11] [36] However, for logarithms that appear in the exponent of a time bound, the base of the logarithm cannot be omitted. For example, O(2 log 2 n) is not the same as O(2 ln n) because the former is equal to O(n) and the latter to O(n 0.6931...). Algorithms with running time O(n log n) are sometimes called linearithmic. [37]
Let be a cyclic group of order , and given ,, and a partition =, let : be the map = {and define maps : and : by (,) = {() + (,) = {+ ()input: a: a generator of G b: an element of G output: An integer x such that a x = b, or failure Initialise i ← 0, a 0 ← 0, b 0 ← 0, x 0 ← 1 ∈ G loop i ← i + 1 x i ← f(x i−1), a i ← g(x i−1, a i−1), b i ← h(x i−1, b i−1) x 2i−1 ← ...
Tables containing common logarithms (base-10) were extensively used in computations prior to the advent of electronic calculators and computers because logarithms convert problems of multiplication and division into much easier addition and subtraction problems. Base-10 logarithms have an additional property that is unique and useful: The ...
A logarithmic unit is a unit that can be used to express a quantity (physical or mathematical) on a logarithmic scale, that is, as being proportional to the value of a logarithm function applied to the ratio of the quantity and a reference quantity of the same type. The choice of unit generally indicates the type of quantity and the base of the ...