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This question already has an answer here:

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.

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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:

$$10^0<7<10^1$$

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

$$10^{1/2}<7<10^1$$

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

$$10^{3/4}<7<10^1$$

And again:

$$10^{3/4}<7<10^{7/8}\\10^{13/16}<7<10^{7/8}$$

etc.

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