Introducing Logarithms

I started logarithms with my (second to bottom) S4 class this week, and I think I managed to introduce it in a way that really helped the students to understand what a logarithm is. First I started the lesson with this recap set of questions on indices.

​As the students were completing it I realised my error in including 4^(1/2) as this can lead to misconceptions that an index of 1/2 is the same as halving. I probed this after the class completed the questions by asking what 9^(1/2) is, and most of them correctly recalled that it was 3.

Next I used the idea from James Tanton's Take on Logs where I wrote some on the board like this
power2(8) = 3
power5(25) = 2
Your turn!
power3(27) = ___
power10(100) = ___

I didn't use the examples from his essay, but rather used ones that linked to the questions from the starter (the answers were still projected), and wrote them from scratch on the whiteboard. Students could easily see the link between the two. As they grew confident I started to use some that were not from the starter. Towards the end I threw in a couple of impossible situations such as power1(73) = ? and power5(-1) = ?. As James suggests, I then made a big deal out of changing power to log, and explaining that we just use a different name for the function. As we went through these examples I stuck to a particular routine. So, for example, for the question power3(27) = the back and forth went like this:​

Me: "Maria, what is the actual question being asked?"
Maria: "3 to the power of what is 27"
Me: "And the answer is?"
Maria: "3"

​I used cold call from Teach Like a Champion 2.0 throughout this process, writing a question up and asking a student. The class came to the phrase "3 to the power of what is 27" as a group rather than me telling them.

​At this point a student asked "Why do we need logs". Fortunately I had this slide from Dr Frost Maths ready, and launched in to talking about how logarithms are the inverse of exponentials and that we need them to solve equations where the unknown is in an index.

​We then did an example problem pair, just to reiterate the process, and again I asked what the question was saying.

Then I set them this exercise that I put together in the style of the ones from the amazing variationtheory.com by Craig Barton. I designed it to try to get students thinking about the connections between each question, though I admit that I still need to work on this aspect of running an exercise like this. However, the questions did bring out some interesting ideas, and many of them were able to spot the impossible ones.

​In going through the answers we once again used Cold Call, and rattled through them pretty quickly, again following the same dialogue as above. That was the end of the first lesson.

In the second lesson I started with a few simple questions as retrieval from the previous day, and then students filled in this stickable from Sarah Hagan, with the slide with information given below.

We then did these excellent place the log on the number line activities from the Mathematics Vision Project.

And finished with students writing their own set of Two Truths and a Lie cards (again from Sarah Hagan) based on logarithms. I collected these in, and in the next lesson we started using these, which I projected through the visualiser. Students then used this ordering activity from Susan Wall.

After this I will be going on to teach the laws of logarithms and solving simple exponential equations using logs.

You can find my folder of resources on this topic here. It includes a work booklet for students, the powerpoints I used, along with some other activities.