Thursday, December 13, 2012

What just MIGHT be (for me) the Secret of Teaching Logarithms

I have been teaching logarithm for a few years now. Each year, no matter how I approach it and how exploratory I make the whole thing to be, I find that my students are fairly unsuccessful at putting everything together, and they always get confused at some point. Last year, I finally had the idea of going back to basic definitions. The whole problem, I think, with kids getting confused with logs all the time is because they simply cannot remember, in the end, what the hell log even means after I make them derive all those rules. So, this year, I started with the definition very firmly, and every time the kids are doing a new problem, I repeat the hell out of that definition until they want to rip me into pieces. And, guess what! I don't care if they want to rip me up. It has worked like a charm. NO ONE is getting confused yet this year by the notion of logs. (I've skipped the exploratory stuff this year, in order to really keep their focus on what's important.)

This is the definition I taught them:

Log is just a way to ask a specific question.
loga(b) asks the question: "What exponent is required to go from a base of a in order to reach a value of b?"

That's IT! We go over that with an example.

For example,
log2(8) means "What exponent is required to go from a base of 2 to reach a value of 8?"
So, log2(8) = ??

The kids said, "3!" (...OK, maybe first they said 4. I cannot remember now. But anyhow, they understood why it would be 3. Either they self-corrected or I corrected them.)

Then, we did some more simple numerical examples, as you always would do before kids start to get confused with logs. In each case, instead of just letting them be robots and following the previous numerical pattern mindlessly, I slowed them down and hammered into them the meaning of log. They had to say it OUT LOUD for every example:

log3(81) means "What exponent is required to go from a base of 3 to a value of 81?" and that's why it's 4.
log5(5)  means "What exponent is required to go from a base of 5 to a value of 5?" and that's why it's 1.
log4(16) means "What exponent is required to go from a base of 4 to a value of 16?" and that's why it's 2.

etc. And then we went over the change of base formula, loga(b) = log(b)/log(a). I am sorry, but I didn't try to make them discover it this year. Derivation is nice if the kids are already getting the basic concept, but else it obfuscates what's already a fairly tricky topic for a majority of kids. We practiced finding some decimal log results using the calculator, and testing them (as exponents) to make sure that they did give approximately the correct values that we desired, starting from the base.

And then we jumped right into solving equations! And the kids did brilliantly. I didn't even make a worksheet, I just started writing things on the board, a couple of simple problems at a time. Each time they got stuck, I just said, "Go back to your definition. What question does log help us ask? How can we use that?"

Each time they worked on a new type of problem and they needed help, they had to laboriously say out loud what the question is that log is asking. "What exponent is required to go from base of ___ to reach a value of ___?" and they then had to identify, based on the equation given, whether that question being posed had already been answered or not. Once they said all of this out loud, they were able to figure out on their own what x was fairly easily, without any help from me.

3x =10  --> "What exponent is required to go from base 3 to reach a value of 10? That hasn't been answered yet." so, log is going to help us ask that of the calculator: log3(10) = x

log4(x) = 3  --> "What exponent is required to go from base 4 to reach a value of x? That has been answered already, 3." So, 43 = x.

logx(36) = 2  --> "What exponent is required to go from base x to reach a value of 36? That has been answered already, 2." So, x2 = 36. For this one, it led us into a brief discussion of why x could not be -6, and of limitations on log inputs.

I was really shocked by how well the kids received this. I even tried after a few problems to introduce to them the memory trick from Amy Gruen, and they looked at me like, "Why would we need this?" (which I can assure you, was not the response I had gotten in the previous year.) I really, truly believe that going back to the definition of logs is the way to teach this often confusing concept.

Shortly after, they were able to do problems such as:

log5(1/5) = ??

log7(7k) = ??  --> "log asks the question, what exponent is required to go from a base of 7 to reach a value of 7k? The answer is, well, k!"

log7(72n-3) = ??

So, being very pleased by their ability to recite and apply log definition, I started to put up some questions of multiple-step equations on the board, again just to let the kids try them first. (They needed a bit of hints only in the beginning, but for the most part they were pretty OK doing them by themselves.)

2*5x =80  --> here was my hint. "Well, log does NOT ask the question, what exponent is required so that when I raise the base of 5 to it AND THEN MULTIPLY BY 2, the final value is 80. So, clearly the 2 here is a bit problematic..." and therefore the kids figured out that it needs to go away first.

-4x =-73 --> here I helped them visualize order of operations by circling the x with the 4, and then circling the negative sign on an outside layer. I use this 7th-grade trick now even with my 11th and 12th graders to help them visualize how to peel away layers of the onion when solving for something.

3*6x - 7 = 20

102x-9 =1098

So, this was all things that happened during our previous class. Today, after they returned, they were still very successful at transferring the log definition onto more complicated equations such as:

 6x = 36x-3     (which I realize, yes, they can easily solve in the future as a "change of base" problem, but since we're on the topic of introducing logs, I just wanted them to see how to apply the log definition to this problem.)

So, this is the question they decided to ask: "What exponent is required in order to go from a base of 6 to a value of  36x-3 ?"  And they decided that the answer to that question has already been provided, as x.

So, log6(36x-3 ) = x

Now they apply a simple log rule of dropping the exponents in the front, which makes:

(x - 3) log6(36) = x

And clearly since they know what log means, they can immediately simplify it now as:
(x - 3)(2) = x

and then just solve the rest as a linear equation. Tada!

Easy breezy. I'm going to always teach logs using definitions from now on. My little logarithm ninjas can even solve exponential equations for x in terms of other variables, and they can also tell me that log6(6m^3) should equal m3. YEAH. Not bad for being only two days into logs, I'd say.

If kids understand the definition of logs as something that asks a certain question, then down the road they won't be so confused when we discuss that 2log2(k) = k, because the log part simply asks the right question, and the rest of the expression actually CARRIES OUT the instruction implied by that question. I find that when the situation looks complicated, I always go back to thinking about the definition of log in my own head. So, I have every reason to be hopeful that my kids, with consistent reinforcement from me, will create the same frame of reference in their little heads.


  1. Hi, I just started reading your blog recently, and I wanted to comment on this because I just finished teaching logs in a pre-calc class for the first time. I had a similar experience where I would try to do harder and harder things with logs and they would keep forgetting the basic definition. I tried going in to log as the inverse of exponential functions, but the way you write log is so visually different from exponential functions, and the connection just didn't get made for a lot of kids.

  2. I did make the connection that log is an inverse operation. I didn't go into the functional nature because we're going on Christmas break after one more class session. I think the right way to introduce that is AFTER the kids are comfortable with the idea of log and exponents being inverse operations, to give them a few exponential functions, ask them to find the inverse functions, and let them discover the pattern that it will always end up being a log function (and vice versa). The functional nature of log is more complex, and so it makes more sense to come later as a general wrapup, I think.

    But, don't worry, Katie... I have taught logs at least 4 or 5 times already and this is the only time I haven't felt like it's totally muddy afterwards for the kids. It's just a difficult topic to teach. I recommend the definition approach next time, even though it sounds really not flashy or exciting. Kids really get it, I think, because they get to focus on just one thing instead of a bunch of little disjointed half-knowledge.

  3. PS. Hi! Sorry, I'm tired but I am not usually this rude. I am always excited to have new readers. :) :) Especially when they leave me messages so I know that they exist!!

  4. Thank you so much for sharing this method! I taught logs to my Algebra 2 class two weeks ago using your definition approach, and I was shocked at how easily all the students were able to understand the concept. This has definitely helped to make me a better teacher. :)

  5. I always used log_a x = y MEANS a^y = x (and color code the letters). I'm thinking the method you're using might be better.

    I'm giving a talk at CAMT 2015 (this year - I'm a bit late reading this post) on teaching logs and exponents. I'm glad I found this - I'll use it (with attribution, of course!).


  6. I came across this blog via Bon@MathFour above posting about this specific blog on logarithms on facebook. I am interested in following this blog but don't use one of the listed newsreaders. Can you assist me in following your blog?

  7. Hi Susan, sorry but I think the newsreaders is the best way. You can also bookmark the page and check back? (Sorry I can't be more helpful in this! I don't do an email subscription thing...)

  8. Hi! I totally agree about needing to get the definition completely automatic. It takes way too much working memory to make sense of the problems otherwise!

    In my search for teaching techniques, I came across this unconventional method that I think is pretty cool, and I'm about to use it with my online students.
    What do you think?

  9. 2^log2(k) = k: is the second 2 in this equation intended to be the base of the log? Or does it mean taking the common log of 2K?