# Calculating Logarithms by repeated multiplication and division?

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?

• Try to find log of 10 or higher using this method. Nov 23, 2022 at 15:24
• 11 is between 100 and 10 so first digit before the decimal point is 1, continuing with a calculator for the exponentiation I get 1.0413. Nov 23, 2022 at 16:42

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.
• Because if $10^a \leq x < 10^{a+1}$, then $a \leq \log_{10} x < a+1$, since $log$ is a monotonic function. Nov 24, 2022 at 13:45