Approximating the Logarithm of Any Base with Continued Fractions

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Continued fractions provide a very effective toolset for approximating functions. Usually the continued fraction expansion of a function approximates the function better than its Taylor or Fourier series. This Demonstration shows the high quality of a continued fraction expansion to approximate the logarithm to an arbitrary real base greater than 1. It uses the Shanks method and is very efficient due to its adaptability for high-speed numerical computer code.

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The logarithm base must be larger than 1, and the number for which the log is computed must be larger than the logarithm base, so must hold.

To make this Demonstration easier to use, the sliders only increment in multiples of 1/10, but Shanks' method is not limited to rationals.

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Contributed by: Andreas Lauschke (March 2011)
Open content licensed under CC BY-NC-SA


Snapshots


Details

To compute the approximation of the logarithm , two sequences of numbers and are computed as follows:

,

,

...

.

Then

.



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