Numeracy and Mathematics Guest Blogs
This week's blog is from Iona Coutts, Numeracy and Mathematics Education Officer with Education Scotland.
Spreading the love of logarithms
Iona Coutts
Education Scotland
Twitter: @onycoutts
March 2021
I have always loved teaching Higher classes, and one of my favourite topics is logarithms. I appreciate that this makes me slightly weird. In recent months, when working with Higher teachers I have asked two questions:
Which Higher topics are most difficult for your learners and
Which is the topic you find hardest to teach?
Logarithms always feature highly in both categories. So in this blog I really want to say that logs are brilliant. Logs are entirely related to bits of maths learners know already. Logs are powerful. And they are Scottish. All of which means that The Love of Logs should be spread unsparingly in Higher Mathematics classrooms! This blog is not meant to be a complete guide to teaching logarithms, rather a series of observations and some suggestions for activities.
A strong start
It's always good to introduce a new topic by illustrating why it is important, and where it fits into the network of knowledge that learners already have. I start with the simple equations opposite. Learners are familiar with powers of two, and some discussion inevitably leads to a mystery power of 4 and a bit. Or even 4 and a big bit as 29 is nearer 32 than 16. With existing knowledge however we are, at present, stumped. What at first glance looks simple, is actually quite tricky and needs some "new" maths.
I would also, at this point, throw in a reminder of the existing knowledge of negative and fractional powers for good measure. They will need to be confident in this throughout the topic and it illustrates that the concept of non-integer powers is not new to them.
I leave them with a dangling carrot - that by then end of this topic, they will be able to solve this innocent looking devil of an equation!
At this point, I would then introduce the graphs of exponential functions. I have come to believe that one of the fundamental strands that needs to be explicitly linked throughout National 5 (and beyond) is that of a locus of Cartesian points that illustrate an algebraic relationship, whether that relationship is linear, quadratic or trigonometric. I think learners see these graphs as separate entities and this link needs to be mentioned at every opportunity. At Higher, we add to this bank of prior knowledge with graphs of exponential and logarithmic functions, polynomials and circles.
One way to reinforce this idea is to go back to a good, old fashioned table of values and a hand-drawn graph for a simple function such as y = 2x. This can really help learners think about the link between the y and x coordinates of points on the curve. They can also explore what happens at the y-axis and to the left of it and relate this to their understanding of indices. It could also yield an approximation to our starting puzzle that improves on a power of "4 and a bit". Once understood, digital tools can then be used really effectively to investigate related graphs obtained through simple translations and a variety of bases.
So far so good, and we haven't touched on logarithms yet. For this, we need to remind learners of their prior understanding of inverse functions, and introduce the idea of the inverse of an exponential function. We are back to our starting examples again.
Some discussion can be had by looking at examples using different bases around reversing an exponential calculation such as those opposite. Using language such as "2 to the power 5 is what?" and "2 to the power what is 64?" gives a familiarity with how these processes are related. This new way of thinking can then be linked to logarithmic notation, and the exponential equations can be written in logarithmic form. Khan Academy has a nice video introduction to logarithms which you can find here.
Some carefully guided practice, such as this activity from Chris McGrane on his excellent startingpointsmaths blog, can help learners become familiar with the notation whilst thinking carefully about what it means.
Once learners are familiar with the relationship between exponential expressions and equations and logarithmic ones, graphs can be revisited and the relationship between the graphs y = ax and y = logax can be explored. At this point, it is interesting to introduce Euler's number, e, and the natural logarithms. Linking to the graph of the derivative and the special property that the derivative of ex (with respect to x) is ex is a nice nod to future learning from Advanced Higher. You can find a great video from Eddie Woo introducing Euler's number here. Learners need some practice at evaluating expressions involving powers of e using their calculator. Solving some simple exponential and logarithmic equations in bases e and 10 can be practised too. It's also worth checking with your learners whose calculators are limited to base e and base 10 logarithms and whose are not, even though I would use calculators as little as possible throughout this topic. It can be helpful to explore some contexts for exponential grown and decay, and a discussion with colleagues from science departments might help with this. The current pandemic has affected so much of our lives, never in my experience has there been a more current context with which to illustrate the impact of exponential growth.
A pause for some history
The mathematics within the Higher course has a history that is rich and colourful, with stories of intrigue and ego, collaboration and cooperation. I think it is important that students know that the maths they are learning was cutting edge at certain points in history and still holds true today, over 400 years later. It's also important to recognise that it continues to be a growing body of knowledge. There is still maths to be done! In the case of John Napier, I think he deserves a mention and a quick chat about his life and legacy. You can find a short biography here and an activity on multiplication using Napier's bones here.
The laws of logarithms
Clearly and explicitly linking the laws of logarithms back to what learners already know (the laws of indices) is really important. The proofs of these laws are fairly accessible and it's valuable to take the time to work through them with learners, but it can help to allow them to play with these new ideas first.
Learners can work through some carefully thought out examples such as comparing
log2 16 + log2 4 and log2 64
log2 32 + log2 2 and log2 64
log10 100 + log10 1000 and log10 100000....
They can try to find a rule and generalise for log2 a + log2 b and then logx a + logx b and try to explain how this links to their prior knowledge. Similar examples can be explored for the other rules.
Having explored these rules, learners can start to use them to simplify expressions and solve equations. You can find a nice series of tasks from Transum here.
By this point, learners have acquired quite a few new skills and it's important from now on that they get regular practice in converting between exponential and logarithmic form, evaluating exponential and logarithmic expressions, simplifying expressions and solving equations using the laws of logarithms. It is desirable for these manipulations to become as fluent as possible.
And of course, now, learners also have all the tools they need to solve our original puzzle. It might not occur to them that they now have a new string to their bow in solving equations. Logarithms are all about finding the power, so if ever they are solving an equation with an unknown power, they can take the logarithm of both sides and apply their log laws from there. Of course, those learners whose calculators can deal with base 2 logarithms could have easily solved this from the get go!!
Applying this knowledge
When all these bases have been covered, learners need to be able to solve problems using these skills. At Higher, these might be linked to evaluating expressions and solving equations such as this question from the 2019 paper.
Initially, questions like these require a significant degree of problem solving and it's amazing what sort of issues can arise from the inclusion of skills learned within the BGE. "Miss, how do I reduce 120 by 15% again?" My advice is to smile patiently and wait. Chances are it will come back to them! It's worth allowing learners the opportunity to try to solve these problems for themselves, maybe in pairs or small groups, rather than teaching them how to do them as if they constitute another skill to be learned. They aren't. They are applications of what they already know.
Talking of applications of what learners already know, the final piece of the jigsaw is representing exponential relationships with straight lines using logarithmic axes. This is an area that I know teachers find difficult, and so I'm going to dedicate another blog just to that. But for now, I'll just say I prefer to view this too as an application of what learners already know rather than an additional skill to be learned, but that's for another day.