# Teaching on topic: logarithms

Fingers crossed.

I arrived in a bad spot.

I do not think I properly lined up my ducks before reaching logarithms. And now I’m sort of stuck.

The book, my text, uses “inverse functions” as the segue. And I’ve walked right up to that line, and realized I would have preferred to approach from another side.

Too late.

Blame the book. Blame the over-packed curriculum. Blame the teacher’s poor planning (well, no, please don’t).

I’m not so wild about teaching at almost the same time as teaching what a log is. But maybe I’m wrong. Could we introduce logs back when we work with exponents in Algebra I? Probably too abstract?

So while I’m not so wild about where I’ve ended up, I’m not sure I have or had a real alternative.

I will use part of the idea from Kate(t)’s neat intro log lesson. And I will make them look like a puzzle. Students like puzzles. And I will separately and carefully get them to graph and its inverse, without clearly making the links. And if all goes well, one kid, then another, will notice the links, the idea will be shared around, and all will be well with the world…

We’ll see. And if I don’t report back…

Good luck! I suppose that for some number of students, it’ll click and then “the earthquake magnitude scale is logarithmic” might help. Hmmn… is the earthquake magnitude scale still logarithmic or only approximately logarithmic now? I know it’s no longer Richter…

Probably too abstract?Likely. Logarithms are the spot for me where the curriculum really shifts to Algebra 2, where the only way to really get at it is an abstraction of an abstraction (and much brain exploding commences).

Here’s a short book on teaching logarithms I found to be a worthwhile read.

A fellow teacher at my school had some success teaching trig functions and inverse trig functions in geometry at the same time. He used a trig table and showed the students how to get the ratio of the length of any 2 sides of a triangle given an angle. Then asked what they could do if they were given the ratio. He gave them a few leading question with familiar values. They picked up immediately. But since it wasn’t part of geometry standards, he only touched on it and didn’t introduce the notation or explored it further.

I imagine the same could be done with a log table. You might also have them build a table using a calculator given a base and a set of exponents. Base of 2 with exponents 0-10 in 0.1 increments. Have students work in groups to reduce tedium? This could also be a good opportunity to show them the table feature in TI calculators along with tblset.

Ask them what is 2^1, 2^2, 2^3.1, 2^5.2 etc. by using only the table.

Proceed with (still using only the table):

2^(some number)=4 what is “some number”?

2^(n)=8 what is n?

If 2^x=4.28709385 what is x? (it’s 2.1, but it’s only a decimal approximation, what if we want an exact value?)

Try with a base of 10. Calculators have that function.

Once they know what they are doing, introduce the notation. Connect with inverse functions and go into the graphs you were planning.

This is one case where a table is more useful to go back and forth and explore instead of using a calculator and have them punch away at the buttons without know what’s happening.

Good luck! Kate(t)’s introductory log lesson looks like a great place to start.

I have tried lots of things to introduce logarithms without finding something that really worked great for me.

One idea that I enjoy and seems to work well for some students is a world population activity. I put the names of countries (all 200 or so that I find in the CIA world factbook) and their populations on little slips of paper. I let each kid draw one (or maybe two or three or five if I think we’ll have time). I get them to work together on making a big long horizontal number line on the board and figuring out what scale to use to graph them all. They see that you can’t tell anything useful from the graph except maybe “India and China have a lot of people, there are a few more countries like the US that have pretty big populations, and then there are a ton of countries that are small”.

Then we make a new axis, labeled on a log scale, but I don’t call it that: I say something like “Let’s talk about how many zeros are at the end of the number, instead of the number itself”. They pretty quickly can figure out where to place 100 million or whatever, but pretty quickly they start to argue about what is halfway between 100 million and a billion. Someone will say 500 million, of course, and then someone will point out that linearly it should really be 550 million, and if you’re lucky a kid will suggest that no, each tick mark is a multiplicative thing, so the halfway point should not be 5, since that would make 25 at the tick mark instead of 10, and …

It’s actually pretty impressive how close to bell-curve the populations look when graphed on a log scale. And you can really appreciate what’s going on, with the big peak around 10-20 million, and see all kinds of interesting stuff that was completely buried in the linear-scale graph. You don’t really lose much, either: you can still see that China and India are really big.

I like the idea from Kate(t)’s lesson that you can teach logarithms almost like a secret code, figuring out the pattern. I don’t like her use of vocabulary, treating “power” as a synonym for “exponent”. To me, in 2^3 = 8, 2 is the base, 3 is the exponent, and 8 is the power. I find that being really careful about those three words is a big help in talking about log and exponent properties. When you write log_b(x) = y, you can say that x is the power and y is the exponent and have it mean something. Maybe you (and Kate) have other ways to handle the vocabulary here.

I really like being able to say a^x * a^y = a^(x+y) and then “when you multiply powers of the same base, you add the exponents”, and then with “a log is an exponent”, I can write log (t*u) = log t + log u and say “when you multiply powers of the same base, you add the exponents”, only now t and u are the powers and log t and log u are the exponents.

Thanks for the discussion/comments. I will be looking here, and at my notes, when I try again.

In my last comment I meant to say “exponential” instead of “logarithmic”. Sorry.

I’ve noticed that students also run into many problems trying to use function notation and understanding functions in general. When I teach algebra, I start the first day by defining what a function is. This typically involves some hand-waving in the direction of relations. By starting with the function definition so early, the students have plenty of time to become comfortable with notation and vocabulary by the time we introduce inverses as an “un-doing” process (usually shortly after systems of equations). I also make sure to point out that “solving” equations is done by applying inverse functions to the equations. I noticed that students who understand how an inverse works will usually approach logarithms more openly, understanding that they can help to solve exponential functions.