How to Calculate Logarithms Without a Calculator

Question

As the title states, I need to be able to calculate logs (base $10$) on paper without a calculator.

For example, how would I calculate $\log(25)$?

Answer 1

This is a method I found a year ago. This method takes a lot of time but it will give an accurate answer.
To calculate log(25):
1) Divide 25 by the nearest power of 10. The condition must be 25 ≥ 10ⁿ.
2) The value of n is 1 because 25 ≥ 10¹. So the initial answer is 1.xxxxxx.
3) Divide 25 by 10¹. The result is 2.5.
4) Raise 2.5 by 10. So 2.5¹⁰ ≈ 9536.7
(Note: The number is raised to 10 because we are already looking for the digits after the decimal point.)
5) For the next values, the same process will be used.
6) Divide 9536.7 by the nearest power of 10.
7) 9536.7 ≥ 10³ so n=3. The answer is now 1.3xxxxx.
8) 9536.7 / 10³ = 9.5367
9) Raise 9.5367 to 10. 9.5367¹⁰ = 6222733625
10) 6222733625 ≥ 10ⁿ so n=9. The answer is now 1.39xxxx.
11) Repeat the same process until you get the desired precision.
12) So log (25) ≈ 1.39794.

This also works on logs with bases other than 10, even with decimals. In solving log_a(x), just replace 10ⁿ with aⁿ. Also in solving for n, simply just divide the number by the base repeatedly until you get a quotient nearest to 1. The number of times you divided is n. (ie. 250/10= 25 (1), 25/10=2.5 (2), so n=2)

This is a method I had formulated on my own so I'm not saying that this is 100% reliable. It would be better to check your answer with a calculator. But I've tried this many times without flaws. No log tables, no need to memorize, just pure math.