Learning Some Logarithms

I have picked up spaced repetition again, so I now have a way to learn some logarithms basically for free, as well as how to use them for mental maths. The values below are good to know and enough to get started (the rest can be roughly interpolated.)

• \(\;\;\;\; \log{1} = 0\)
• \(\;\;\;\; \log{2} \approx 0.3\)
• \(\;\;\;\; \log{3} \approx 0.5\)
• \(\;\;\;\; \log{5} \approx 0.7\)
• \(\;\;\;\; \log{8} \approx 0.9\)
• \(\;\;\;\; \log{10} = 1\)

A few weeks after I memorised that batch, I added a second batch to fill out some of the gaps. These are probably overkill but convenient once the memories are formed.

• \(\;\;\;\; (\log{1.25} \approx 0.1)\)
• \(\;\;\;\; (\log{1.6} \approx 0.2)\)
• \(\;\;\;\; (\log{2.5} \approx 0.4)\)
• \(\;\;\;\; (\log{4} \approx 0.6)\)
• \(\;\;\;\; (\log{6} \approx 0.8)\)
• \(\;\;\;\; (\log{7} \approx 0.85)\)
• \(\;\;\;\; (\log{9} \approx 0.95)\)

Base ten logarithms are particularly convenient because we write numbers in base ten, so we only need to know the logarithms of 1–10 to compute the logarithm of any other number.

For larger numbers, we can approximate using the rule that turns multiplication into addition² This rule was in fact why Napier invented logarithms in the first place! Multiplication problems are hairy but addition is easier.

\[\log{50} = \log{(5 \times 10)} = \log{5} + \log{10} \approx 0.7 + 1 = 1.7\]

(Incorrect by 0.06 %.)

We can always just round to the nearest number with a bunch of zeroes on it:

\[\log{87234} \approx \log{(8.7 \times 10^4)} = \log{8.7} + \log{10^4} \approx 0.93 + 4 = 4.93\]

(Incorrect by 0.2 %.)

To approximate log 8.7, we see that it is about a third of the way between 8 and 10, and since all continuous functions are linear when you look up close, we just go a third of the way from log 8 to log 10, both of which we know!

Smaller numbers work on the same principle:

\[\log{0.055} = \log{(5.5 \times 10^{-2})} = \log{5.5} + \log{10^{-2}} \approx 0.73 - 2 = -1.27\]

(Incorrect by 0.8 %.)

from Hacker News https://ift.tt/J4PxQRn