Logarithms are a notoriously tricky topic for A-level students to grasp. Having just about got their heads around laws of indices, we suddenly expect them to blithely apply a new – seemingly arbitrary – list of rules that turn index rules literally on their heads:
Some people argue that logarithms are an outdated topic and have no place in our curriculum. They were useful back in the day, but smartphones have all but made calculators obsolete, let alone slide rules and log tables. What need have we for this confusingly different mathematics?
The key idea underpinning logarithms is this: They are a way of counting multiplicatively. And there are, in fact, many good reasons for wanting to count in this way, rather than in the traditional additive sense.
I asked my year 12 class how many Christmas trees were sold in the US each year. One guessed “300 million” and another guessed (I suspect facetiously) “3”. Given that the true number is around 30 million, which of these students was closer to the truth? And if you went with the 300 million guy (who overestimated by 270 million) rather than the guy with the ludicrous underestimate (29,999,997 too low) then presumably you are using a different scale to gauge accuracy than a straightforward additive one. Consciously or not, we usually prefer a logarithmic scale for such comparisons, especially when the values vary considerably in size. The first guess was 10 times too big, but the second guess was 10 million times too small.
Without thinking too hard, where do you think the number 1,000 belongs on this number line?
While not many people would have put it right in the middle, most people’s initial reaction is somewhat further along than an additive scale would place it. The true position is somewhere inside the blob above ‘1’ on an additive scale. But isn’t there some sense in which a million is ‘halfway’ from a thousand to a billion? Yes – a multiplicative sense.
Dave gets a 20k pay rise. Is he much better off? It depends on what he was on before, doesn’t it? On the one hand, 20k is £20k, but £520k is only 4% bigger than £500k, while £24k is a 500% increase on £4k. In fact, whenever we invoke proportion, we are dealing with a logarithmic number sense.
With this new form of number sense, we need new rules for moving along the number line, and that’s where thinking about numbers from the point of view of the exponent starts to really pay off.
If you haven’t yet seen the superb ViHart video on logarithms, I highly recommend it. It makes sense of this multiplicative number scale beautifully.
When you start to look around you, there is evidence of a multiplicative number sense everywhere.
- How can going down from 2 cigarettes to 1 be harder than from 20 to 19?
- In what universe is having a 6th kid less of a big deal than having a 2nd?
- How could a 10th friend joining a party less noteworthy than the 3rd one?
- Who drives 10 minutes to save £5 off a £15 item, but not a £150 one?
Given that logarithms and this logarithmic number sense are such a natural part of how our brains interpret quantities, it makes perfect sense to try and find a rigorous mathematical way to represent and manipulate them.
Once the ideas behind logarithms make more sense, the intuition behind the log rules starts to become clearer:
The next step is to develop notation. We need to provide students with a convenient way to ‘read’ the logarithm notation: “What power do I need, with this given base, to result in this number?”
And once we’re familiar with this, knowing which bases are most commonly used and why is always helpful:
And solving real problems early is always helpful, I find. Compound interest problems should be well within the capacity of A-level students, but as soon as the unknown value is the length of time rather than the interest rate or the amounts, it becomes a problem of trial and error… or logarithms. It’s worth explicitly demonstrating the distinction between problems involving powers that we know (that are usually solved by rooting) and those involving powers that we don’t know (that need this whole new function to solve):
The ‘number of digits’ interpretation of logarithms is a particularly helpful one. While log10 isn’t exactly the same, if you round it up it gives you exactly that. A really helpful idea from the excellent new Underground Mathematics website involves working out just how far up an exponential graph will go: https://undergroundmathematics.org/exp-and-log/reach-for-the-stars. We need to grasp the fact that, as fast as exponential graphs increase, logarithms do the opposite, requiring us to travel literally miles along the horizontal axis just to reach a reasonable height on the y-axis (for the record, 500 miles along gives you about 1 handspan in height on a log10 graph).
As usual, I’d welcome any feedback / ideas / corrections on the ideas & resources included.
Just remember, next time you come second in a pub quiz tie-breaker, to check whether your answer was multiplicatively closest. Although you should bear in mind that, until our newly enlightened year 12s start administering them, “because of logarithms” is an argument unlikely to carry much weight…