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The logarithm in the table, however, is of that sine value divided by 10,000,000. [1]: p. 19 The logarithm is again presented as an integer with an implied denominator of 10,000,000. The table consists of 45 pairs of facing pages. Each pair is labeled at the top with an angle, from 0 to 44 degrees, and at the bottom from 90 to 45 degrees.
It states that under appropriate conditions the logarithm of a summation is essentially equal to the logarithm of the maximum term in the summation. These conditions are (see also proof below) that (1) the number of terms in the sum is large and (2) the terms themselves scale exponentially with this number.
In computational complexity theory, a log-space reduction is a reduction computable by a deterministic Turing machine using logarithmic space.Conceptually, this means it can keep a constant number of pointers into the input, along with a logarithmic number of fixed-size integers. [1]
This definition gives rise to a function that coincides with the binary logarithm on the powers of two, [3] but it is different for other integers, giving the 2-adic order rather than the logarithm. [4] The modern form of a binary logarithm, applying to any number (not just powers of two) was considered explicitly by Leonhard Euler in 1739 ...
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:
Logarithmic spiral (pitch 10°) A section of the Mandelbrot set following a logarithmic spiral. A logarithmic spiral, equiangular spiral, or growth spiral is a self-similar spiral curve that often appears in nature. The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line" ("ewige Linie").
In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function Li s (z) of order s and argument z.Only for special values of s does the polylogarithm reduce to an elementary function such as the natural logarithm or a rational function.
Stochastic logarithm is an inverse operation to stochastic exponential: If , then (()) =.Conversely, if and , then (()) = /. [1]Unlike the natural logarithm (), which depends only of the value of at time , the stochastic logarithm () depends not only on but on the whole history of in the time interval [,].