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This reduces the problem to one where the argument of the logarithm is in a restricted range, the interval [1, 2), simplifying the second step of computing the fractional part (the mantissa of the logarithm). For any x > 0, there exists a unique integer n such that 2 n ≤ x < 2 n+1, or equivalently 1 ≤ 2 −n x < 2.
For example, with z = 1.5 the third approximation yields 0.4167, which is about 0.011 greater than ln(1.5) = 0.405465, and the ninth approximation yields 0.40553, which is only about 0.0001 greater. The n th partial sum can approximate ln( z ) with arbitrary precision, provided the number of summands n is large enough.
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 = 9,808,357 + 0.09543. We can then get 10 9,808,357 × 10 0.09543 ≈ 1.25 × 10 9,808,357. Similarly, factorials can be approximated by summing the logarithms of the ...
If p is a probability, then p/(1 − p) is the corresponding odds; the logit of the probability is the logarithm of the odds, i.e.: = = = = (). The base of the logarithm function used is of little importance in the present article, as long as it is greater than 1, but the natural logarithm with base e is the one most often used.
The Pennsylvania parents of a 21-year-old blind and deaf man with cerebral palsy who died in September after being starved for months have been charged in connection to his death, authorities ...
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[2] A familiar example of logarithmic growth is a number, N , in positional notation , which grows as log b ( N ), where b is the base of the number system used, e.g. 10 for decimal arithmetic. [ 3 ]