Logarithm

In mathematics, a logarithm is a function that gives the in the equation bn = x. It is usually written as logb x = n. For example:

Algorithm

To calculate logb(x) if b and x are rational numbers and x ≥ b > 1:

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If n0 is the largest natural number such that bn0 ≤ x or, alternately,

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: n_0 = lfloor log_b(x) floor

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then

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: log_b(x) = n_0 + rac{1}{log_{x / b^{n_0}}(b)}

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This algorithm recursively produces the continued fraction

Related Topics:
Algorithm - Recursively - Continued fraction

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: log_b(x) = n_0 + rac{1}{n_1 + rac{1}{n_2 + rac{1}{n_3 + cdots}}}.

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The logarithms produced are irrational for most inputs.

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To use irrational numbers as inputs, apply the algorithm to successively detailed rational approximations. The limit of the result series should converge to the actual result.

Related Topics:
Irrational number - Series

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:ln 2 = 2sum_{n = 0}^{infty }rac{(1/5)^{

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2,n+1}}{ 2,n+1}+2sum_{n = 0}^{infty }rac{(1/7)^{

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2,n+1}}{ 2,n+1}

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