The standard explanation is this, “a logarithm is an exponent”.
This is true. But I’m not sure it’s particularly helpful for a student who is struggling. I have been burning a lot of mental calories over the past few versions of College Algebra trying to come up with ways into logarithms that will have more explanatory power and be more intuitively inviting to my students.
I realize the quest may be quixotic.
And I understand that this is well-trodden ground.
But consider this equation:
In the standard interpretation, this asks, What exponent do I put on 2 to get 9?
What if instead, we thought about it this way: How many factors of 2 are in 9?
More than 3, fewer than 4.
One has no factors of 2 in it, so:
But even better,
because the expression on the left asks how many factors of 2 are in AB while the one on the right asks how many are in A and how many are in B, then adds them. Some of the 2s are in A, the rest are in B.
The language is problematic, I know. The answer to the question How many factors of 2 are there? is properly “2”.
But we use the language of factor in our exponential work, so it may not be too problematic in this context (and few College Algebra students are number theorists).
And of course I’m not naive enough to expect that this will solve all of our logarithm difficulties. But I’ve got something to work with.
Thoughts and critiques?
Overthinking my teaching 
First I’d like to say Hi Christopher! From Kelly (Rivette) Darke:) I have been really enjoying your blog posts. Regarding this post, at first glance it seems like a useful attempt to clarify some connections between the two operations/functions of exponentiation and logarithms. I like the question: “How many factors?” versus “What exponent…” In my opinion, the language of “What exponent…” always is so confusing. Your suggested language is much more intuitive in my opinion. One thing I did notice with your take on the equation log2(AB)=Log2(A)+log2(B) is that the base of 2 (2 being prime) helps with the decomposition of the idea you are explaining above. If the base is 10 ( or any nonprime) the factors will not always be counted in such a clear way since both A and B could contain proper factors of 10. Perhaps that is what you meant by: “The language is problematic, I know. The answer to the question How many factors of 2 are there? is properly “2″. ” I was confused by that sentence. That’s all for now and keep up the fabulous work. You keep me thinking (and overthinking) every day!
I like this. I was trying to think of what a student would say to, “How many factors of 2 are in 1.5?” Wouldn’t they say none?
I think we really need to have students on board for the easy ones: how many factors of 2 are in 4, in 8, in 64? etc. Before we can begin this game. And then introduce others as very interesting puzzles: How many in 7? How many in 9? There’s something analogous to remainders here. I wonder if leveraging that idea is helpful.. and then there are puzzles such as, How many in 1/2? How many in 1/4? … How many in 1? is certainly a puzzle. How many in -2?
Figuring out which puzzles you want them to make contact with and in what order is certainly an interesting question.
I think I’m realizing just now how much understanding of the numberline (and operations on it) one needs to understand logs.
We started logs yesterday, and I used some of jd’s ideas. I know I also got some my ideas from Kate Nowak. There was someone else in on a discussion a while back of using L() or P() instead of log(), but I can’t remember who. One of my students commented on how fun it was.
I like your idea of using the word factor. I always have trouble with how many times I have to use the word ‘to’. “What power do we have to raise 2 to to get 32?” I definitely say power instead of exponent.
I’m not taken with your approach, though I certainly appreciate the effort. I was working with a precalculus student I tutor (my son is also in precalculus, but his school is about half a chapter behind that of my student, though both use the same old Larson-Hostetler stand-by text except that my son’s has preliminary sections that my student’s does not) on logarithms last night. Since the sections on logs follow a section on exponential functions, most of the material (though not all of it) seemed to make sense to the student in the context of the function-inverse function relationship. In exponential functions, we take a base to a power and get a result. In logs, we need to figure out what power to take the base to get a given result. And frankly, losing that fact seems to me to lose the sense of what logs are: namely the exponent needed to get a given result starting from a certain base.
Seems to me that it’s better to work within that framework in various ways to help students make connections between what they already know about how addition, multiplication, and exponents work (or perhaps what they SHOULD know: if they don’t, this is another chance to learn the interrelationships). Sooner or later, the comfortable “multiplication IS repeated addition” and “exponentiation IS repeated multiplication” is going to break down, because the fact is that once you get past integers (or maybe even whole numbers) it’s really stretching a point to insist (as many teachers do) that the analogy holds. However are students to make sense of the notion of irrational powers using the repeated multiplication myth?
Have you tried applying your language/model to the laws of logarithms? If so, how well do they seem to hold up there? I certainly don’t quibble with the idea that many students are ill-equipped to deal with, much less make sense of, logs, but I’m inclined to think that the problem has more to do with how poorly they’re grounded in previous key ideas than that there’s something fundamentally difficult about logarithms.
Picking up where bwfrank left off, the model I build with students is a number line type. Two lines actually, one just above the other. The top is the power scale and the bottom is the logarithm (or exponent) scale. One on the power scale is directly above zero on the exponent scale. If we are looking at a base-10 scale, then 1000 on the power scale is above 3 on the exponent scale and so on. The unit on the power scale is powers of ten and the unit on the exponent scale is one.
The payoff is that a logarithm is a distance.
Log base-10 of 1000 is the distance from zero on the exponent scale. 3 units. If I provide students with a more detailed power scale, one that has 2 in the right place, then by direct measurement we can find log base-10 of 2 equal about .3 units from 0 on the exponent scale.
We use this model to develop the log properties and demonstrate the proportionality of different bases.
Seeing logarithms as exponents happens along the way but the students have something more concrete to hang their hats on.
To all of you…thank you so, so much for engaging here. I am blessed to have so many smart people willing to read, and then to write back.
Next post will up the ante on my we need new ways to talk about logs argument. In the meantime, some thoughts on your thoughts…
Kelly, my friend, thanks for stopping by to say ‘hi’. It has been way too long. Perhaps my language has led you astray of my meaning (which argues against this particular language; I’ll think about that). If we’re looking at log, base 10, of 200, we can still split that up as (log, base 10, of 20) + (log, base 10, of 10). The “number of factors of 10” here refers to the number of times 10 appears as a factor, and is supposed to be analogous to the division question, How many 5’s are 100? The primeness of 2 in my example is irrelevant. And I totally get how using the language of factor counting is going to be an impediment.
Brian, Aaron and Sue, thanks for the added input. I’ll definitely think about the number line business, and of course I appreciate Kate Nowak’s inductive approach. Mine wouldn’t supplant something like what she has in mind; it is intended to provide an additional way of talking about logarithms that gets us out of the a logarithm is an exponent rut.
And MPG, I do not in any way intend to replace the definition of logs. That’s the key to the whole business. It’s awkward notation and we need to go back and forth between exponential form and logarithmic form many, many times until it becomes a part of our soul if we have any hopes of making any sense of logarithms. I think we’re in agreement on that.
Where I’m working is on the concept image side of things. If we can accept that a certain percentage of our students in a college algebra/Alg II/precalc/whatever course are going to be working from their concept image of logarithms a substantial amount of the time (which I believe to be an irrefutable premise), then what concept image can we encourage that will be helpful? My experience tells me that a logarithm is an exponent is not a productive concept image for lots of students. It is technically true, but not generative for many.
In that spirit, I’m not sure I’m so troubled by a concept image that relies on exponentiation as repeated multiplication. At the entry level, exponentiation is repeated multiplication, just as at the entry level, multiplication is repeated addition. And these are powerful concept images. They are not robust enough to generalize to the real numbers in either case, of course, but they are robust enough to (1) get us started in the topics, (2) lean on for sense making later on, and (3) help us bump up against important questions, including the question, “when does our image break down?”
One place it breaks down is when the exponent is irrational (which is almost always with logarithms). This issue of irrational exponents seems to be a really big one for lots of folks. I have to admit that I don’t get its importance. I get its importance in Mathematics, of course. But I don’t get its importance in mathematics below Calc (or even arguably undergrad analysis, typically a junior-year course mostly for math majors). For people who are going to use mathematics to do things other than study more mathematics, I don’t get what’s so toxic about them understanding that 2^sqrt(2) is approximately equal to 2^1.414 (which it is).
And I’m not certain that even most calculus students have sorted out these issues for multiplication, never mind for exponentiation. Calc students have the fraction thing down pat, but do they understand the trouble that is introduced by talking about 4*sqrt(2)?
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