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I am curious as to how the original log tables were generated. How did they determine that log base 10 of 7 were approximately 0.845...

I've seen various hand calculations of square roots and sine and come up with my own and I have found they help me understand the nature of those functions. I was wondering if I could strength my intuitions of logs beyond the simple notion of what exponent raises the number to the correct value.


marked as duplicate by Henning Makholm, Tanner Swett, Trevor Gunn, mfl, Simply Beautiful Art Jul 11 '17 at 23:20

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Here's one simple solution using bisection:

Assume that it is known how to calculate $b^x$ for any $x$, or at least for any rational $x$. Then, note that:


Thus, we know that $\log_{10}(7)$ is between $0$ and $1$. We can also calculate $10^{1/2}$ and we know that


We then "bisect" the interval $[1/2,1]$ and find that


And again:




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