One of the topics that confuses many of my students is logarithms. That’s a shame because once you understand them logarithms can be very easy – the exam questions are often gift marks. But if you don’t ‘get’ them then logs seem like bizarre black magic.
So in the next few posts I’ll talk a bit about logs and try to explain them and help you to understand them. Before getting on to the what and the how I’ll talk about the ‘why’.
Why Logarithms?
So why do we use logarithms? What were they invented for, apart from annoying students?
Logs were invented to do complex sums before the invention of calculators. Imagine something like 468 x 793. OK, it’s easy enough to do by hand as a long multiplication – but it’ll take a few minutes and there’s a good chance of making a mistake. Or what about finding the square root of 7396. You can do it using trial and error and a clever algorithm but it’s a real pain.
Now imagine you have to do dozens of these calculations every day…
Enter logarithms. They turn multiplication into addition, which is much simpler. And square root just becomes diving by two! Here’s how it works. Don’t bother about the ‘becomes’ and ‘gives’ for now, just notice how much simpler it is than the original sum:
468 x 793 = 37133018 = 371124 becomes log(46826) + log(793) = 2.6702 + 2.8993 = 5.5695 which gives 371107
Not totally accurate because of the rounded decimals – but a very good approximation.
Similarly:
√7396 = 86 becomes log(7396) ÷ 2 = 3.8690 ÷ 2 = 1.9345 which gives 86.00031
If you remember that logs were designed to turn multiplication into addition etc it’ll help to understand what’s going on.
So how about those magic ‘becomes’ and ‘gives’ steps, how were they done before calculators? Well, you simply looked the numbers up in a book of ‘log tables’ – a list of numbers and their logs. I’m old enough to remember using these ‘four figure tables’ in my exams! For an even quicker but less accurate answer you could also use a slide rule, which basically did the same thing.
Today of course we have calculators – so why do we still use logs?
Logarithms Today
With most of us having calculators that can do these sums instantly, so what’s the point of learning logs? Well it turns out that lots of things in the real world follow a logarithmic pattern and logs/exponentials are a very useful way of modelling them. I’ll talk more about that in the section on natural logs.