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The geometric mean can also be expressed as the exponential of the arithmetic mean of logarithms. [4] By using logarithmic identities to transform the formula, the multiplications can be expressed as a sum and the power as a multiplication:
The power mean could be generalized further to the generalized f-mean: (, …,) = (= ()) This covers the geometric mean without using a limit with f(x) = log(x). The power mean is obtained for f(x) = x p. Properties of these means are studied in de Carvalho (2016).
In wireless communication, "the local-mean power expressed in logarithmic values, such as dB or neper, has a normal (i.e., Gaussian) distribution." [90] Also, the random obstruction of radio signals due to large buildings and hills, called shadowing, is often modeled as a log-normal distribution.
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. Derivations also use the log definitions x = b log b (x ...
In general, logarithms can be calculated using power series or the arithmetic–geometric mean, or be retrieved from a precalculated logarithm table that provides a fixed precision. [ 45 ] [ 46 ] Newton's method , an iterative method to solve equations approximately, can also be used to calculate the logarithm, because its inverse function, the ...
The computational complexity of computing the natural logarithm using the arithmetic-geometric mean (for both of the above methods) is (() ). Here, n is the number of digits of precision at which the natural logarithm is to be evaluated, and M ( n ) is the computational complexity of multiplying two n -digit numbers.
The logarithmic mean of two numbers is smaller than the arithmetic mean and the generalized mean with exponent greater than 1. However, it is larger than the geometric mean and the harmonic mean, respectively. The inequalities are strict unless both numbers are equal.
The arithmetic mean, or less precisely the average, of a list of n numbers x 1, x 2, . . . , x n is the sum of the numbers divided by n: + + +. The geometric mean is similar, except that it is only defined for a list of nonnegative real numbers, and uses multiplication and a root in place of addition and division: