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.
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
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.