Calculating ln Natural Logarithms

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...

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Math has never been my thing, but since I’ve been reading a lot of useful posts here on the forum, and since you guys make it easy to understand, I am beginning to like it :slight_smile:

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In the previous post I took 81, since it is an exact exponent of 3.

What if this is not the case?

Well, we need a small extra step then.

Let’s take ln 30.
27 = 3^3 \approx {2.7}^{3.3}

Observe that in this post and the previous one, if we subtract 10% from the base, we can add about 10% to the exponent.

30 is about 11% more than 27, So add 0.11 to 3.3 to get 3.41.

Mathematically:
ln \, 1.11 \approx 0.11

More general:
ln \, 1+x \approx x if and only if x is small. How small? Preferably less than 10%.

In the case of ln 1.11, the actual answer is more like 0.104.

However, for memorization, take the same number (and learn how to adjust later).

So if we know the natural log for a number, adding 1% to the number gives an increase of 0.01 in the answer.

Ln 100 = 4.605
Ln 101 = 4.615

This makes it easy to start with a number, guess the ln of it and move closer to the number we actually want to take the ln of

Back to our example.
ln 30 \approx 3.41

Don’t forget the 1.0026 factor. See my previous post.
We need to subtract 2.6 ‰.

For calculating 341 * 26, I start with 34 * 26, since this is easy and expand the answer afterwards:

34 * 26 = 30^2 - 4^2 = 900 - 16 = 884
340 * 26 = 8840
341 * 26 = 8840 + 26 = 8866

The complement of 8866 is 1134, so
3.41 - 0.008866 = 3.40 + 0.001134 = 3.401134 and this is our mentally calculated logarithm of 30.

To show how accurate this answer is:
e^{3.401134} = 29,9981...
Pretty close to 30.

When doing this mentally I take a couple of shortcuts, but this is already a long text…

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