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In mathematics, for given real numbers a and b, the logarithm log b a is a number x such that b x = a.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.
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 ...
The open (closed) upper half-space is the half-space of all (x 1, x 2, ..., x n) such that x n > 0 (≥ 0). The open (closed) lower half-space is defined similarly, by requiring that x n be negative (non-positive). A half-space may be specified by a linear inequality, derived from the linear equation that specifies the defining hyperplane. A ...
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.
Semi-log plot of the Internet host count over time shown on a logarithmic scale. A logarithmic scale (or log scale) is a method used to display numerical data that spans a broad range of values, especially when there are significant differences between the magnitudes of the numbers involved.
The natural logarithm of x is generally written as ln x, log e x, or sometimes, if the base e is implicit, simply log x. [2] [3] Parentheses are sometimes added for clarity, giving ln(x), log e (x), or log(x). This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity.
Euler's formula is ubiquitous in mathematics, physics, chemistry, and engineering. The physicist Richard Feynman called the equation "our jewel" and "the most remarkable formula in mathematics". [2] When x = π, Euler's formula may be rewritten as e iπ + 1 = 0 or e iπ = −1, which is known as Euler's identity.
A log–log plot of y = x (blue), y = x 2 (green), and y = x 3 (red). Note the logarithmic scale markings on each of the axes, and that the log x and log y axes (where the logarithms are 0) are where x and y themselves are 1. Comparison of linear, concave, and convex functions when plotted using a linear scale (left) or a log scale (right).