Calculating ln Natural Logarithms

I’ll show calculating ln \, 81 with an example:

e \approx 2.7 \approx 3

3 of course is almost 10% off, but let’s take 3 for the first number.

81 = 9^2 = 3^4

So log_3 \, 81 = 4

Next step; moving from 3 to 2.7

3 divided by 2.7 = 1.1 and ln\,1.1 \approx 1.1.

There is a reason why this is and I’ll make a separate post about this.

So log_{2.7} \, 81 =4 * 1.1 = 4.4

We are getting close.

We now need to subtract a factor of 1.00126 and this factor is constant!

Subtracting a factor of 1.00126 can be calculated as subtracting 1.26 ‰.

This a tenth of 1.26%.

I usually take 1.25‰ = 5/4‰ which is oftentimes so much easier to calculate and by then we are already in very accurate territory.

I’ll show both.

Take 4.4 and multiply by 1.25 = adding a quarter = 5.5.

Divide by 1,000 = 0.0055.

Subtract 0.0055 from 4.4 = 4.3945.

To show how accurate this is, let’s show the 1.26‰ subtraction from to 4.4

4.4 - 4.4 * 0.00126 = 4.394456

Compare with the actual number of ln\,81 = 4.39444915...