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The logarithm in the table, however, is of that sine value divided by 10,000,000. [1]: p. 19 The logarithm is again presented as an integer with an implied denominator of 10,000,000. The table consists of 45 pairs of facing pages. Each pair is labeled at the top with an angle, from 0 to 44 degrees, and at the bottom from 90 to 45 degrees.
In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the 3 rd power: 1000 = 10 3 = 10 × 10 × 10.
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.
Here we employ a method called "indirect expansion" to expand the given function. This method uses the known Taylor expansion of the exponential function. In order to expand (1 + x)e x as a Taylor series in x, we use the known Taylor series of function e x:
Legendre polynomials are also useful in expanding functions of the form (this is the same as before, written a little differently): + = = (), which arise naturally in multipole expansions. The left-hand side of the equation is the generating function for the Legendre polynomials.
The logarithm keys (log for base-10 and ln for base-e) on a typical scientific calculator. The advent of hand-held calculators largely eliminated the use of common logarithms as an aid to computation. The numerical value for logarithm to the base 10 can be calculated with the following identities: [5]
The roots of the Iwasawa logarithm log p (z) are exactly the elements of C p of the form p r ·ζ where r is a rational number and ζ is a root of unity. [4] Note that there is no analogue in C p of Euler's identity, e 2πi = 1. This is a corollary of Strassmann's theorem.
Steps of the Pohlig–Hellman algorithm. In group theory, the Pohlig–Hellman algorithm, sometimes credited as the Silver–Pohlig–Hellman algorithm, [1] is a special-purpose algorithm for computing discrete logarithms in a finite abelian group whose order is a smooth integer.