The 4.3 comes from log(e) = 0,4342944819.

Most people don’t know where ‘e’ comes from. e comes from a series of interest payments, like this:

If I put one dollar in the bank and I get 100% interest, after one year I have 2 dollars.

I can ask myself how much money I have if I get interest payed out twice a year instead of once a year.

Intuitively a lot of people think it does not matter wheter we get twice 50% or once 100%.

But they forget the compounding effect of interest on interest.

Let’s do the calculation. If I get 100% per year I get 50% per half year.

So after half a year I get 50% on 1 dollar = 1 dollar 50 cents and after the second half year I get 50% on the 1

dollar 50 = 75 cent, making a total of 2.25.

The calculation is:

(1+1/2)^2

If I do this for 4 quarters the calculation becomes:

(1+1/4)^4

You might see a pattern here:

If I get paid n times, the calculation becomes:

(1+1/n)^n

If n becomes bigger and bigger, the result for this calculation tends to go to ‘e’:

1 2.0000

2 2.2500

4 2.4414

10 2.5937

50 2.6916

100 2.7048

1,000 2.7169

10,000 2.7181

100,000 2.7183

1,000,000 2.7183

The log of this number then also tends to go to ‘43’:

1 2.0000 0.3010

2 2.2500 0.3522

4 2.4414 0.3876

10 2.5937 0.4139

50 2.6916 0.4300

100 2.7048 0.4321

1,000 2.7169 0.4341

10,000 2.7181 0.4343

100,000 2.7183 0.4343

1,000,000 2.7183 0.4343

Around 50 the number is 0.43.

It means that for differences around 2% we can take 0.43.

For the smallest of differences the number is just 1% bigger, 0.4343.

For big differences like 10%, the number is slightly smaller, 0.41…

For practical purposes I usually take 42.

42 is divisible by 2, 3, 6, and 7 and this makes it very handy for mental calculation.

An example will make this clear:

Let’s say you memorized log(60) and want to calculate log(61):

61 = 60 X (1+1/60) therefore

log(61) = log(60) + log(1+1/60)

Using ‘42’:

log(1+1/60) ~= 42/60 ~= 7/10 ~= .7

Using ‘43’:

log(1+1/60) ~= 43/60 ~= 7/10 + 1/60 ~= .717

It is a long story to explain.

However, mentally it goes like this:

61 is 1/60 more than 60. 42/60 = 7 (forgetting about the decimal point).

Now set the decimal point:

1/60 is a little over 1%, so my answer should be a little lower than 1%.

‘7’ should be 0.007.

Then I realise that for numbers smaller than 2% I should take 43 instead of 42.

So I add 1/60 to 7 to get to ‘717’ or 0.00717 with the decimal point.