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As a consequence, log b (x) diverges to infinity (gets bigger than any given number) if x grows to infinity, provided that b is greater than one. In that case, log b (x) is an increasing function. For b < 1, log b (x) tends to minus infinity instead. When x approaches zero, log b x goes to minus infinity for b > 1 (plus infinity for b < 1 ...
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 mathematical notation for using the common logarithm is log(x), [4] log 10 (x), [5] or sometimes Log(x) with a capital L; [a] on calculators, it is printed as "log", but mathematicians usually mean natural logarithm (logarithm with base e ≈ 2.71828) rather than common logarithm when writing "log".
In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable X is log-normally distributed, then Y = ln( X ) has a normal distribution.
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.
As the common log of ten is one, of a hundred is two, and a thousand is three, the concept of common logarithms is very close to the decimal-positional number system. The common log is said to have base 10, but base 10,000 is ancient and still common in East Asia.
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).
Every family of sets with n different sets has at least log 2 n elements in its union, with equality when the family is a power set. [30] Every partial cube with n vertices has isometric dimension at least log 2 n, and has at most 1 / 2 n log 2 n edges, with equality when the partial cube is a hypercube graph. [31]