So we calculated the ln’s of 81 and 30 in a separate thread.

For smaller numbers - like 2 - the following background story is interesting. At least it is for me!

It might become a long story, so I may have to split this into separate posts.

An interesting analysis is this.

If I have a savings account that pays 100% per year, then I double my money in a year.

If I start with 1 I end up with 2.

Now, If instead I ask for two half year payments of 50%, I end up with more.

My 1 now becomes 2.25.

(After the first payment my 1 becomes 1.5 and making 50% on 1.5 makes it 2.25.)

I could ask for quarterly payments of 25%. Still a 100% on a years basis.

Now my 1 becomes 2.44…

This increase does not go on forever and becomes smaller and smaller with each increase of the number of payments.

If I take the this exercise to continuous payment, I get this:

\\lim_{n \to \infty} \,\,\, (1 + p/n)^n = e^{\,p}

Let that sink in for a moment, since we will be using this.

If I use 1 for p then I can calculate e.

Or at least come close to it.

Let’s do a series of n = 1, 2, 3, …

n=1:

(1+1/1)^1 = 2

n=2:

(1+1/2)^2 = 2.25

n=3:

(1+1/3)^3 = 2.37…

n=4:

(1+1/4)^4 = 1.5625^2 = 2.44…

Let’s skip some steps.

n=8:

(1+1/8)^8 = 2.56…

Btw, you calculate this mentally in 3 steps:

1: 1.125^2 = 1.266…

A simple way to calculate this is to assume you have $1000 in a savings account giving you 12.5%. After the first payment you have $1125. the next payment gives you again $125, making your new total $1250. However you also receive 12.5% over the $125 you got from the first payment.

This is about $16, making you new total $1,266.

2: 1.266^2 = 1.60…

There is an easy way to calculate this.

Start with 1,266 * 1.25.

This is adding a quarter of 1266 to 1266.

1266 / 4 = 300 +15 + 1.5 = 316,50

Add 316,50 to 1266 and get 1,582.5

Now add 0.016 times 1,266 to 1,582,50.

So take 1.6% of 1,266.

1% is 12,66 and 0.6% is about 7.2, making it about 19.86.

We need 17.50 to go from 1582,50 to 1600.

So adding 19.86 takes the total to just over 1600.

Round to 1600 and putting the decimal point back gives 1.6.

3: 1.6^2 = 2.56.

See that with each step we get closer to e \approx 2.71?

If you can do these calculations, you are well on your way to getting at least a feel for ln(2), which I will tell about in future posts.