I just finished a unit of my Open University Maths degree that’s all about exponents and logarithms (unit 13 of MU123). This is the penultimate unit and these final units are where the difficulty has started to stack up for me.
The 2020 COVID pandemic was when all of us had lots of exposure to the idea and reality of exponential curves. In fact, it seemed early on like the people who really understood what exponential growth can do were the most worried. Cancer’s another thing where something that keeps on doubling or increasing can cause real havoc when it compounds. The book talks a bit about Moore’s law, though I recall watching a talk at JuliaCon in 2018 where Sophie Wilson spoke about all the ways where we may be edging up at various physical realities these days that might slow things down.
Logarithms were the other big part of the module, and these were fascinating to work with. I had familiarity with the idea of them, mainly just as a button on a calculator, but it was really interesting to start to play around with all the ways they could be useful. Prior to the widespread availability of calculators they were the de facto way of doing big calculations, since using something like logarithm tables or a slide rule would give you a practically useful approximation of your answer in a way that didn’t require you to do complicated calculations in your head.
I got hold of both an old copy of a book of logarithm tables as well as a slide rule — old tech ftw — and look forward to becoming more familiar with them once the unit has come to an end in a few weeks.
The previous unit (trigonometry) touched on working with radians and (doing some exercises) it was immediately clear why someone would want to use radians instead of degrees as the unit when working with a certain kind of problem. Similarly, for this exponents & logarithms unit, we came across e, Euler’s number, and had some exposure to how there were quite a few places where it made sense to work with e as the base of our logarithms instead of base 10. I really enjoy these parts of mathematics, where abstract concepts or things that people come up with allow us to manipulate ideas and symbols and objects that in turn allow us to solve problems, or think about things in new ways. In many ways these are the things I enjoy the most while studying the course materials.
We came up on a few places where we were asked to prove something using various identities that we’d previously explored. For example, we prove that for any base b and any positive numbers x and y, that:
\[\log_b x - \log_b y = \log_b(\frac{x}{y})\]
We’re not given a great deal of guidance on this task aside from an analogous example. I know the idea of ‘proof’ is a big thing in higher-level mathematics so I’m looking forward to getting a bit more experience with this.
As with most of the previous units, the bigger picture is somewhat elusive. I know a lot more about exponents and logarithms than I knew a few weeks ago, but I’m still unsure of how it connects to everything else around it. I’m hoping that comes with time, but in the meanwhile I’m trying to write these blogposts to at least take a step back and think through those bigger connections.
In the end, logarithms and exponents are another trick, another tool in the box, and another example of how mental models or abstractions can allow for other new thoughts to be had. We end up using some of the index laws that we learned a few months back to allow us to manipulate these new symbols. Manipulating symbols allows us to either simplify things (so we can think at a different level) or move further into something more complex (so we can see what’s going on, through whatever lens we’ve brought to look through).
As a project and as a path to be explored, it’s exciting to take these steps even if sometimes you just have to trust that everything will come together in the end. Next up: “Mathematics Everywhere”, the final unit which is all about practical applications using the things we’ve learned earlier in the module, with a serving of abstract mathematics thrown in for good measure!