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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 ...
Because log(x) is the sum of the terms of the form log(1 + 2 −k) corresponding to those k for which the factor 1 + 2 −k was included in the product P, log(x) may be computed by simple addition, using a table of log(1 + 2 −k) for all k. Any base may be used for the logarithm table. [53]
[1]: p. 35 The proportion along the top rows, starting with 1 is 0.99. The entries in each column are in proportion 0.9995. (Note that 0.9995 = 1-1/2000, allowing "tolerably easy" multiplication by halving, shifting and subtracting.) Napier uses the first column to computing the logarithm of .99, using log of .9995, which he already has.
The log semiring has applications in mathematical optimization, since it replaces the non-smooth maximum and minimum by a smooth operation. The log semiring also arises when working with numbers that are logarithms (measured on a logarithmic scale), such as decibels (see Decibel § Addition), log probability, or log-likelihoods.
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".
The main arithmetic operations are addition, subtraction, multiplication, and division. Arithmetic is an elementary branch of mathematics that studies numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms.
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