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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.
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]
However, in general settings, the logarithm tends to be a multi-valued function. For example, the complex logarithm is the multi-valued inverse of the complex exponential function. Similarly, the discrete logarithm is the multi-valued inverse of the exponential function in finite groups; it has uses in public-key cryptography.
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 arithmetic modulo an integer m , the more commonly used term is index : One can write k = ind b a (mod m ) (read "the index of a to the base b modulo m ") for b k ≡ a (mod m ) if b is a primitive ...
Every concave function that is nonnegative on its domain is log-concave. However, the reverse does not necessarily hold. An example is the Gaussian function f(x) = exp(−x 2 /2) which is log-concave since log f(x) = −x 2 /2 is a concave function of x. But f is not concave since the second derivative is positive for | x | > 1:
where B is the incomplete beta function. A Poisson compounded with Log( p )-distributed random variables has a negative binomial distribution . In other words, if N is a random variable with a Poisson distribution , and X i , i = 1, 2, 3, ... is an infinite sequence of independent identically distributed random variables each having a Log( p ...
The following is a list of integrals (antiderivative functions) of logarithmic functions. For a complete list of integral functions, see list of integrals. Note: x > 0 is assumed throughout this article, and the constant of integration is omitted for simplicity.
Graph of log 2 x as a function of a positive real number x. In mathematics, the binary logarithm (log 2 n) is the power to which the number 2 must be raised to obtain the value n. That is, for any real number x, = =.
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