We can use logarithm to substitute multiplication and exponentiation to addition and multiplication respectively. But in order to get the original value, we have to use antilogarithm. Is there a way to calculate antilogarithm mentally?
Yes, there is.
I wrote a long post about how to calculate the antilog of 0.1, 0.2, 0.3, etc here:
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Now, let’s say you need to know the antilog of 1.21.
10^{1.21} =
10^1 * 10^{0.2} * 10^{0.01} =
10 * 10^{0.2} * 10^{0.01}
Using the previous post we get 10^{.2} = 1.6 minus correction.
As the correction we subtracted .97%.
For simplicity, let’s say we subtracted 1%. Remember this 1%.
The last part is calculating 10^.01
2^{10} = 1024 and log 2 = 0.30103...
If log 2 = 0.30103, then log (2^{10}) = 10 * log 2 = 3.0103...
If we round this number to 3.01, we get log 1024 = 3.01
Then log 1.024 = 3.01 - 3 = 0.01
In other words 10^{0.01} = 1.024
So just add 2.4 %.
So to get 10^{1.21} we take 10 * 1.6, minus 1%, plus 2.4% = 16 plus 1.4% = 16.224
(16*14 = 224).
A calculator gives: 16.218.
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Doing it this way we can calculate antilogarithms to 2 digits.
The 3rd digit is easy to add.
10^{0.001} = 1.0023... or just add 0.23%.
From the previous post:
We mentally calculated 10^{1.21} to be 16.224.
If we now want to calculate: 10^{1.214} we take 16.224 and add 4 * 0.23% = 0.92%.
If we need a quick, rough estimate, we round 0.92% to 1% and get: 16.224 + 0.16 = 16.384.
If we need a better estimate, we round 0.92% down to 0.9% and get: 16.224 + 0.16 - 0.016 = 16.224 + 0.144 = 16.368.
10^{1.214} = 16.368 using a calculator. This is the exact same, because we over calculated in the previous post and under calculated here.
After this the system does not change anymore.
4th digit:
10^0.0001 =1.00023… or just add 0.023%.
5th digit:
10^0.00001 =1.000023… or just add 0.0023%.
So now you can calculate antilogs with any amount of digits.