Another look at “Logarithms!”

OK, even I will admit that the logarithm rules do not seem to be infused with joy.  But once you know them, there is a way to apply them that my physics students seem to get a kick out of.  It’s a little exercise I learned by reading Professor David Mermin’s essay, “Logarithms!” in his excellent book, “Boojums All the Way Through” (Cambridge University Press, 1990).

The goal of the exercise is to be able to determine common (base-10) logarithms without a calculator.  At the time Mermin wrote, scientific calculators were actually not so common.  Now, everyone has access to them and Google will calculate them for you as well.  But that’s not the point.  It feels almost magical to be able to do this stuff without a calculator.  I would say it’s a hoot, definitely worth of the exclamation point Professor Mermin put in the title of his original article.

Here’s the basic plan.  Start by observing some coincidences, some “almost” equalities involving powers of small prime numbers and small powers of 10. Then use the logarithm rules to deduce the values of the logarithms.

[All of the examples that follow are from Mermin.]

For example, it happens that…

210=1024 and 103=1000

So we would be almost correct saying 210   = 103.

Then we could take the log of both sides, applying the power rule for logs to get:

10log 2 = log 1000

but we know that log 1000 = 3.

So 10log 2 = 3 and that means log 2 = 3/10 =.3

I have never presented this to a class without at least a handful of kids grabbing their calculators to see if we are close.  Their calculators tell them that log 2 = .3010 and I hear “Whoaaa!” (but I teach an impressively nerdy group).

Now what about log 3?  We need another coincidence, another “almost” equality.

While it is true that 32=9 which is almost 10, we can come closer with

34 = 81 which is almost 80.  This is helpful because 80 is also 10×2.

Starting from 34 = 10 × 23

we can take the log of both sides and actually enjoy ourselves as we apply the log rules to get:

4 log 3 = 1 + 3 log 2

And we already know log 2 — it’s (roughly) .3, which we can substitute in to get

4 log 3 = 1 + 3 × .3 = 1.9

log 3 = 1.9 ÷ 4 = .475

And when check out log 3 on your calculator, you see .4771….whoaaa.

We can get log 4 immediately from the fact that

Log 4 = log 22 = 2 log 2  ≈ 2 × .3 = .6  (whoaaa)

and log 5 = log (10 ÷ 2) = log 10 – log 2 ≈ 1 – .3 = .7 (Can I get another whoaaa?)

OK, I think you see the flavor of how this works now.  We just need to keep finding helpful coincidences…

Like 72 = 49 ≈ 50 = 100 ÷ 2.  Or  112 = 121 ≈ 120 = 10 × 2× 3.

These starting points expand our collection of approximate logs.  But you have to know your log rules to build the collection.

Mermin does not stop here.  He writes:

“Those whom the schools have taught to hate numbers will probably want to stop after this preliminary warmup, but the real fun is only beginning.”

He then goes on to improve on all of these estimated logs by constructing systems of equations built on closer, dazzling coincidences.  I recommend the essay and in fact the whole collection.