# How to calculate antilogarithm mentally

#1

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?

#2

Yes, there is.

I wrote a long post about how to calculate the antilog of 0.1, 0.2, 0.3, etc here:

#3

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 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.

#4

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.