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Ok, so I found this thing where they say you can find the digits of the common logarithm of a number n by taking n^10, seeing which two powers of ten the result is between and noting the lower one, then dividing by that lower power of ten to get the next number to raise to the tenth power for calculating the next digit. This is the website: https://calipertimepieces.com/2017/12/calculating-logarithms-manually/

I did it for the first 2 digits of 2, 3, and 5 and it seemed to work. So I thought, huh, does this work for any base. So I try calculating log_6 (2). I do 2^6 etc, and I get .215, which is not the first 3 digits of log_6 (2)... in base 10, it is the first 3 senary digits of log_6 (2).

So I'm wondering if this is just a weird fluke or if it works iff the positional and log bases are the same and if it does work I would love to know why it would work. My question is: What are the mathematics behind this method of calculating logarithms?

Daniel Schwartz
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This works, but only for the base that you're writing numbers in - so it will give you the decimal digits for $\log_{10}$, the binary digits for $\log_2$, and so on.

The reason is because $\log x^a = a \log x$, so $\log_{10} x^{10} = 10 \log_{10} x$ so it's just the same thing as extracting the decimal digits of a number by repeatedly multiplying by 10 and taking the floor.

ConMan
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  • Ok, but how does looking at where the 10th power of a number n is between the powers of the base tell you the digit of the log? Am I just failing to make an obvious connection? – Daniel Schwartz Nov 24 '22 at 13:08
  • Because if $10^a \leq x < 10^{a+1}$, then $a \leq \log_{10} x < a+1$, since $log$ is a monotonic function. – ConMan Nov 24 '22 at 13:45