We talked about mentally calculating logarithms. Now, let’s go the other way.
How does one mentally calculate 100.1, 100.2, etc.?
You could of course memorize the numbers, but this is the mental calculation subforum, so we need to calculate them. Mentally. Ouch.
Think about it; how would you calculate 100.1?
It is - by definition - the tenth root of 10, so one option is to try a number and raise that to the tenth power - for example by repeatedly squaring. And then iteratively refine the initial guess.
This takes time. It is great training of course, but let’s find an easier way.
We know that log(2) = 0.301. 0.30103 if you want to be precise.
Since this number is very close to 0.3, we could use this number and make a small correction for the fact that we are working with 0.301 instead of 0.3.
If we do this we can calculate these numbers with about 4 digits precision.
As always, it is a lot more work to explain than to do, so bear with me.
If log(2) = 0.301, this means that 100.3 is almost 2.
If we do the correction later, we can start by stating that 100.3 = 2.
Then, 100.6 = 4 and 100.9 = 8.
Now, from the nine numbers 0.1 - 0.9, we have already 3 numbers covered.
Let’s continue adding 0.3:
101.2 = 16,
101.5 = 32,
101.8 = 64,
102.1 = 128,
102.4 = 256,
102.7 = 512.
That was not difficult, right?
Let’s bring the exponent under 1:
If 101.2 = 16, then 100.2 = 1.6
If 101.5 = 32, then 100.5 = 3.2
If 101.8 = 64, then 100.8 = 6.4
If 102.1 = 128, then 100.1 = 1.28
If 102.4 = 256, then 100.4 = 2.56
If 102.7 = 512, then 100.7 = 5.12
You probably see where this is going.
To finish the series we can add:
100.0 = 1
101.0 = 10
Putting them all in order we get:
100.0 = 1
100.1 = 1.28
100.2 = 1.6
100.3 = 2
100.4 = 2.56
100.5 = 3.2
100.6 = 4
100.7 = 5.12
100.8 = 6.4
100.9 = 8
101.0 = 10
Take a moment to see what has happened here. We calculated all these numbers by adding 0.3 on one side and doubling the last number on the other side.
This can be done in a matter of seconds.
If you want to know 100.4, all you need to know is ‘what multiple of 3 ends in a 4?’
Well; 3 x 8=24.
So, we take .3 x 8 = 2.4:
10 ( .3 * 8 ) = 10 2.4 = 2 8 = 256 => 10.4 = 2.56
That version is the long version. The short version goes like this:
What multiple of 3 ends in 4? Well, 24 (8 times 3) => use 8.
28 = 256, so 2.56.
Done.
Try this out for yourself with a couple of numbers before we move to the calculation of the necessary correction next.
The correction.
An example.
10.3 = 2
Multiple here was 1, so we subtract .24% once.
2 - .24% = 2 - .0048 = 1.9952
Another one:
10.2 = 1.6. Multiple is 4. We subtract 4 times 0.24%.
.24 * 4 =.96.
1.6 - .96% = 1.6 - 0.01536 = 1.58464.
Btw. This might seem like a difficult calculation, but instead of 1.6, I take 160, then subtract 1% or 1.6 = 158.4.
Finally I realize that 1% is 0.04% too much.
Since 4 * 16 = 64, we need to add 64, move it a couple of places to the right, nd add it to 158.4 to get 158.464.
The general rule is this. We subtract ‘the multiple’ times 0.24% from the calculated number.
How accurate is this? Well, here are the numbers:
Mentally calculated
10.1 = 1.2585
10.2 = 1.58464
10.3 = 1.9952
10.4 = 2.51085
10.5 = 3.1616
10.6 = 3.9808
10.7 = 5.00941
10.8 = 6.30784
10.9 = 7.9424
Actual
10.1 = 1.25893
10.2 = 1.58489
10.3 = 1.99526
10.4 = 2.51189
10.5 = 3.16228
10.6 = 3.98107
10.7 = 5.01187
10.8 = 6.30957
10.9 = 7.94328
