Quick calculations with a few logarithms
If you can remember a few logarithms, you can do many calculations quite easily without the aid of calculators or computers.
Try to remember the logarithms of just seven numbers:
Log 2 = 0.30, log 3 = 0.48, log 7 = 0.85, log 11= 1.04, log 13= 1.11, log 17 = 1.23 and log 19=1.28.
The logarithm of a composite number is equal to the sum of the logarithms of its prime factors; you can formulate the following table of logarithms:
Number Logarithm Number Logarithm
2 0.30 11 1.04
3 0.48 12 1.08
4 0.60 13 1.11
5 0.70 14 1.15
6 0.78 15 1.18
7 0.85 16 1.20
8 0.90 17 1.23
9 0.95 18 1.26
19 1.28
Note: Logarithm of 11 is 1.04. It means log 1.1 = 0.04. Log 5 = log (10/2) = log10 – log 2.
With the above figures in hand you can do a lot of calculations quite easily.
1. What is 2 raised to power 37?
Logarithm of 2 is 0.30.
37 times 0.30 is 11.10
Since logarithm of 1.2 is 0.08 and logarithm of 1.3 is 0.11, antilog of 0.10 is about 1.26.
So antilog of 11.10 is 1.26 * 10^11.
That is 2^37 = 1.26 * 10^11.
2. What is the 31st root of a 35 digit number?
This can be answered even without knowing the number.
Log of 35th digit number is 34. …
Log of 31st root = 34. …./ 31
This lies between 34/31 and 34.99/31 or between 1.09 and 1.13.
From the above table, we can see that in this interval, logarithm of 13 is 1.11.
So, 31st root of a 35 digit number is 13.
3. What is the 64th root of a 20 digit number?
Answer is 2.
4. What is the cube root of say 74567?
Log of 74567 = 4.873 (interpolating between log 7 =0.85 and log 8 = 0.90).
4.873 / 3 = 1.624.
Antilog of 0.624 is 4.24 (because log 4 = 0.60 and log 5 = 0.70).
So, cube root of 74567 is about 42.4
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