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If the underlying group is cyclic of order, by substituting as and noting that two powers are equal if and only if the exponents are equivalent modulo the order of the base, in this case modulo , we get that is one of the solutions of the equation () = ().
Each relation contributes one equation to a system of linear equations in r unknowns, namely the discrete logarithms of the r primes in the factor base. This stage is embarrassingly parallel and easy to divide among many computers. The second stage solves the system of linear equations to compute the discrete logs of the factor base.
The identities of logarithms can be used to approximate large numbers. Note that log b (a) + log b (c) = log b (ac), where a, b, and c are arbitrary constants. 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 ...
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For any number a in this list, one can compute log 10 a. For example, log 10 10000 = 4, and log 10 0.001 = −3. These are instances of the discrete logarithm problem. Other base-10 logarithms in the real numbers are not instances of the discrete logarithm problem, because they involve non-integer exponents. For example, the equation log 10 53 ...
Microsoft Math contains features that are designed to assist in solving mathematics, science, and tech-related problems, as well as to educate the user. The application features such tools as a graphing calculator and a unit converter. It also includes a triangle solver and an equation solver that provides step-by-step solutions to each problem.
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.
An easy way to calculate log 2 n on calculators that do not have a log 2 function is to use the natural logarithm (ln) or the common logarithm (log or log 10) functions, which are found on most scientific calculators. To change the logarithm base to 2 from e, 10, or any other base b, one can use the formulae: [50] [53]