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Another example is the p-adic logarithm, the inverse function of the p-adic exponential. Both are defined via Taylor series analogous to the real case. [98] In the context of differential geometry, the exponential map maps the tangent space at a point of a manifold to a neighborhood of that point. Its inverse is also called the logarithmic (or ...
Finally, in an operation too simple to really be called a fourth stage, the results of the second and third stages can be rearranged by simple algebraic manipulation to work out the desired discrete logarithm x = f 0 log g (−1) + f 1 log g 2 + f 2 log g 3 + ··· + f r log g p r − s.
Legendre's constant is a mathematical constant occurring in a formula constructed by Adrien-Marie Legendre to approximate the behavior of the prime-counting function (). The value that corresponds precisely to its asymptotic behavior is now known to be 1.
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 = 1.724276… means that 10 1.724276… = 53.
Here, and are the two bases we will be using for the logarithms. They cannot be 1, because the logarithm function is not well defined for the base of 1. [citation needed] The number will be what the logarithm is evaluating, so it must be a positive number.
In a series of articles published between 1870 and 1885, Ernst Meissel described (and used) a practical combinatorial way of evaluating π(x): Let p 1, p 2,…, p n be the first n primes and denote by Φ(m,n) the number of natural numbers not greater than m which are divisible by none of the p i for any i ≤ n. Then