This week I have been mostly thinking about natural logarithms.
I'm not sure if there's any practical purpose to this in everyday life, but it means expressing a number as a certain power of e.
e (or e^1) = 2.7 1828 1828 459 (I have put the spaces in because this is how you memorise it to 12 decimal places). The natural log of e = 1.
e^2 = 7.389, so the natural log of 7.389 = 2.0, and so on.
To approximate the natural log of any number up to 1,000, you just need to memorise two tables, I've rounded all the numbers a bit because this is only an approximation:
Table 1
Natural log of 2.7 = 1
Natural log of 7.4 = 2
Natural log of 20 = 3
Natural log of 55 = 4
Natural log of 148 = 5
Natural log of 403 = 6
Table 2
50% = 0.41
100% = 0.69
150% = 0.92
You can memorise more increments if you wish, but it's just as easy interpolating between them.
So... how do you work out the natural log of, for example 80?
The highest number from Table 1 that is lower than 80 is 55, so the first part of your answer is 4.
Then subtract 55 from 80 = 25. That's 'about' 50% of 55. You look in Table 2, and the next part of your answer is 'about' 0.41. You scale that down accordingly i.e. take (25 ÷ 27.5) x 0.41, so knock off 'a bit less than one-tenth' of 0.41, which is 0.38.
Add the two together, the natural log of 80 is 4.38.
Check on calculator, yup, the answer is 4.382.
Natural log of 100 is a bit trickier.
The first part of the answer is also 4, from Table 1.
100 - 55 = 45, which is a bit more than 80%, so you have to interpolate between 0.41 and 0.69 from Table 2. Take three-fifths of 0.28 (i.e. 0.69 - 0.41) and round it up, which is 0.19, and add that to 0.41, and the second part of the answer is = 0.6.
Add 4 + 0.6, the natural log of 100 is 4.6.
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After half an hour's practice, you can get most to the nearest decimal place, two decimal places if you're lucky, and it's a bit more challenging than playing Candy Crush or something.
Debate Magazine
