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In mathematics, change of base can mean any of several things: Changing numeral bases, such as converting from base 2 to base 10 . This is known as base conversion. The logarithmic change-of-base formula, one of the logarithmic identities used frequently in algebra and calculus.
The identities of logarithms can be used to approximate large numbers. Note that log b (a) + log b (c) = log b (ac), where a, b, and c are arbitrary constants. Suppose that one wants to approximate the 44th Mersenne prime, 2 32,582,657 −1. To get the base-10 logarithm, we would multiply 32,582,657 by log 10 (2), getting 9,808,357.09543 ...
In mathematics, the logarithm to base b is the inverse function of exponentiation with base b. That means that the logarithm of a number x to the base b is the exponent to which b must be raised to produce x. For example, since 1000 = 10 3, the logarithm base of 1000 is 3, or log 10 (1000) = 3.
Because logarithms in different bases differ from each other only by a constant factor, algorithms that run in O(log 2 n) time can also be said to run in, say, O(log 13 n) time. The base of the logarithm in expressions such as O(log n) or O(n log n) is therefore not important and can be omitted.
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".
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 ...
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 number theory, the more commonly used term is index: we can write x = ind r a (mod m) (read "the index of a to the base r modulo m") for r x ≡ a (mod m) if r is a primitive root of m and gcd(a,m) = 1.
This change can be computed by substituting the "old" coordinates for their expressions in terms of the "new" coordinates. More precisely, if f(x) is the expression of the function in terms of the old coordinates, and if x = Ay is the change-of-base formula, then f(Ay) is the expression of the same function in terms of the new coordinates.