#commonlog

One person (ADEINSATILZEDA0TIL9 on Discord) correctly guessed that the continuation of the pattern was 10, 4π, 16, 20, 8π ... but couldn’t spot the general pattern.  Which is fair, as I say this is both an approximation and a part of maths that... doesn’t actually crop up much in pure maths, and is more of a science/engineering thing!

The pattern drops out by taking the common logarithm (log to the base 10) of each of these terms, and rounding to two decimal places.  The results (all logs in this post to base 10, all figures correct to 2dp): log 2 = 0.30 log 2½ = 0.40 log π = 0.50 log 4 = 0.60 log 5 = 0.70 log 2π = 0.80 log 8 = 0.90

And of course

log 10 = 1.00 log 4π = 1.10 log 16 = 1.20 log 20 = 1.30 log 8π = 1.40

... though the pattern breaks down at log 32 = 1.51 as the errors start to build up!

Some of these are remarkable in how close an approximation they are!  Common log is something I was hardly taught in my maths classes (let’s face it, we only brought logs up at all so that we could use natural log in calculus) and there was something very joyful about spotting these approximate patterns.  I was spurred to research this from watching 3blue1brown’s “Lockdown Math” series, in particular Grant’s video on logarithm fundamentals: https://www.youtube.com/watch?v=cEvgcoyZvB4 log 2 is the heart of all this.  Since 2^10 is so close to 10^3, it does make sense that the common log of 2 would be very close to 3/10.  With 9dp, it reads 0.301029995... exceptionally close to 0.30103 to 5dp.  Which got me thinking about what taking the common log of successive powers of 2 might look like!  Here are the first 33 powers of 2 to 5 decimal places:

The pattern holds throughout (though past here multiples of three having 3 digits confuses things): the logs of all these successive powers of 2 traces out the three times table!  The first two digits (including the one before the decimal point) are the multiple of 3, the next two are the number itself, and the next two are the multiple of 3 again!  If you can remember your three times table up to 33 x 3 = 99, then you have a quick way to remember the common log of powers of 2 all the way up to 2^33!

It’s even fun to notice things like there being a certain number of multiples of three starting with a certain digit corresponds to the number of powers of 2 with a certain number of digits: there are three two-digit multiples of 3 starting with 2, so there are three three-digit powers of 2.  There are four three-digit multiples of 3 starting with 3 (for the sake of argument here, 1 x 3 = 03), so there are four powers of 2 with four digits!