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https://en.wikipedia.org/wiki/Logarithm

Feynman's algorithm

While at Los Alamos National Laboratory working on the Manhattan Project, Richard Feynman developed a bit-processing algorithm that is similar to long division and was later used in the Connection Machine. The algorithm uses the fact that every real number \(\displaystyle{ 1 < x < 2}\) is representable as a product of distinct factors of the form \(\displaystyle{ 1 + 2^{−k}}\). The algorithm sequentially builds that product \(\displaystyle{ P}\): if \(\displaystyle{ P \cdot (1 + 2^{−k}) < x}\), then it changes \(\displaystyle{ P}\) to \(\displaystyle{ P \cdot (1 + 2^{−k})}\). It then increases \(\displaystyle{ k}\) by one regardless. The algorithm stops when \(\displaystyle{ k}\) is large enough to give the desired accuracy. Because \(\displaystyle{ \log(x)}\) is the sum of the terms of the form \(\displaystyle{ \log(1 + 2^{−k})}\) corresponding to those \(\displaystyle{ k}\) for which the factor \(\displaystyle{ 1 + 2^{−k}}\) was included in the product \(\displaystyle{ P}\), \(\displaystyle{ \log(x)}\) may be computed by simple addition, using a table of \(\displaystyle{ \log(1 + 2^{−k})}\) for all \(\displaystyle{ k}\). Any base may be used for the logarithm table.

https://math.stackexchange.com/questions/879241/feynmans-algorithm-for-computing-a-logarithm-of-a-number-in-1-2