Question

How can I find log* of any number (say log 2 ) without using calculator, log table, or without memorising any log value of 2, 3, 5 etc.

* log x = log_{​​​​10}​​x

Answer 1

I will explain it with an example.
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 value.
12) So log (25) ≈ 1.39794.


This process is also complex and will take a lot of time.