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Demonstrating log* 4 = 2 for the base-e iterated logarithm. The value of the iterated logarithm can be found by "zig-zagging" on the curve y = log b (x) from the input n, to the interval [0,1]. In this case, b = e. The zig-zagging entails starting from the point (n, 0) and iteratively moving to (n, log b (n) ), to (0, log b (n) ), to (log b (n ...
The LSE function is often encountered when the usual arithmetic computations are performed on a logarithmic scale, as in log probability. [5]Similar to multiplication operations in linear-scale becoming simple additions in log-scale, an addition operation in linear-scale becomes the LSE in log-scale:
In this section, we present the general case of the Pohlig–Hellman algorithm. The core ingredients are the algorithm from the previous section (to compute a logarithm modulo each prime power in the group order) and the Chinese remainder theorem (to combine these to a logarithm in the full group).
One part of this machine called an "endless spindle" allowed the mechanical expression of the relation = (+), [14] with the aim of extracting the logarithm of a sum as a sum of logarithms. A LNS has been used in the Gravity Pipe special-purpose supercomputer [15] that won the Gordon Bell Prize in 1999.
The elementary functions are constructed by composing arithmetic operations, the exponential function (), the natural logarithm (), trigonometric functions (,), and their inverses. The complexity of an elementary function is equivalent to that of its inverse, since all elementary functions are analytic and hence invertible by means of Newton's ...
From the mean value theorem, there exists a value ξ in the interval between x and y where the derivative f ′ equals the slope of the secant line: (,): ′ = ()The logarithmic mean is obtained as the value of ξ by substituting ln for f and similarly for its corresponding derivative:
Pollard's rho algorithm for logarithms is an algorithm introduced by John Pollard in 1978 to solve the discrete logarithm problem, ...
The logarithm of a product is the sum of the logarithms of the numbers being multiplied; the logarithm of the ratio of two numbers is the difference of the logarithms. The logarithm of the p-th power of a number is p times the logarithm of the number itself; the logarithm of a p-th root is the logarithm of the number divided by p. The following ...