Do students confuse $\log_ab$ and $\log a^b$?

Question

I recently observed a group of students being introduced to logarithms for the first time. Some of them had trouble writing $\log_ab$ properly, and it looked more like $\log a^b$. All logarithms have a base, so $\log a^b$ doesn't even mean anything (the natural logarithm in that context was denoted by $\ln$ and base 10 by $\lg$), but that might not be obvious to the students.

This made me wonder: Do students think that there is an $a^b$ in the notation $\log_ab$? If yes, how does this manifest itself? I'm mainly looking for actual teaching experiences.

My concern is that logarithm is related to exponentiation and there is something that looks like exponentiation in the notation, and students could make false connections. In particular the students might turn the rule $\log_ab^r=r\log_ab$ into $\log_ab=b\log a$ or something of the kind. It would be better to use parentheses to separate $\log_a(b)$ from $\log(a^b)$ (or even $\log_{a^b}(\cdot)$), but the textbooks do not use parentheses with logarithms and trigonometric functions unless the input is long and students have to cope with that.

Another concern is that the logarithm is defined so that $$ \log_ab=x\iff a^x=b, $$ but the "$a^b$ in the logarithm" might make students think that there ought to be an $a^b$ in the exponential equation. This could turn the definition into $$ \log_ab=x\iff a^b=x $$ in the students' heads. (I know this second definition makes no sense, but that rarely keeps students from doing things.)

Are these concerns unnecessary? If you have found that the logarithm notation indeed confuses students, how have you been able to help them?

Answer 1

I don't believe students subconsciously misread the notation as an exponent with a word in front. I think they are not sure at all how to read the notation!

When using logarithms (whether in a basic, introductory algebra course, or an advanced vector calculus course) I always use parentheses to distinguish the argument of the operation, e.g. $\log_a(b)$. But still, there are always plenty of students who (i) don't know how to read that expression verbally, (ii) don't know how to interpret it conceptually and, consequently, from both, (iii) have no idea how to evaluate or manipulate such an expression.

I try to stress the exponential aspect, introducing examples like $\log_2(8)$ and reading it out loud as, "To what exponent would we need to raise 2 to have 8 as an outcome?" Sometimes this helps. But I even worry that the grammar of that verbal reading of the expression confuses some students just as much! I mean, is this any better: "What exponent do we need to raise 2 to, to get 8 out?" I think not. (And the fact that I picked 2 as the base here only makes it worse; the confusion is just as bad with other bases that don't sound like "to".)

What's even more shocking to me is that, every semester, I find students "solving a problem" and getting an answer that's something like $x=\frac{4}{\log}$. I wish I were joking! But no, their confusion is so extensive that they don't see anything wrong with such an expression, with an "empty" log like that. I typically respond by saying, "This is like writing $x=3+\sqrt{}$ with nothing underneath; it has no meaning." But I so rarely get to follow up with those particular students, so I can't be sure whether that comment helps them better understand the concept.

So, I wouldn't say your concerns are unnecessary; rather, I think the root problem(s) precede confusion about super/subscripts. Below, I list a few things that I believe may be related to this confusion, and try to make some suggestions about how to address them.

1. Problem: Poor arithmetic skills, especially with basic exponentiation. If a student doesn't recognize immediately that, say, $81=3^4$, then they're not going to see any sense in the expression $\log_3(81)=4$. It will look like black magic to them. Addressing this: Take the time to show that $3^4=3\cdot 3\cdot 3\cdot 3=81$ when you introduce this example. It will take more time in class, but it will help prevent some from getting lost right off the bat.
2. Problem: Difficulty understanding the notion of an inverse operation, or being able to ask/answer a question that probes this concept. Look at the sentences I used above: "To what exponent should we raise ... to get ...?" This seems like a simple question to teachers like us who can move "forward and backward" with operations. But to a student, they're used to evaluating expressions forward, plugging and chugging. This may be the first time (except for square roots) that they face an operation that asks them to "undo" something. Addressing this: Perhaps it's best to introduce the fundamental idea of an inverse operation before tackling exponentials and logarithms. Most College Algebra textbooks I've worked with tend to have Inverses as part of their chapter about exponents/logs, but perhaps we can do a better job of synthesizing all those ideas and presenting them as one.
3. Problem: Perhaps $\log_a(b)$ is just genuinely bad notation. I hesitate to say this because I don't have a good proposal for a replacement. But after several years witnessing the same conceptual and procedural errors from students, I truly wonder whether the notation itself is part of the problem. Our temptation is to say something like, "Well, it's just the way we write down the concept; if they understood what it meant underneath, they would understand the notation." Or, we might say, "Well, that's how it's written everywhere; we can't change it now." But I just don't know. Hopefully, someone else can link to some actual research about this idea. Addressing this problem: I don't know. Does anyone have a good proposal for a new notation and reasonable evidence for why it's better for students learning the concept?

Answer 2

When I learned about logarithm, my math teacher used a rather unconventional notation to prevent us from doing such mistakes: instead of writing down $\log_ab$ or $\log_a(b)$, he insisted on writing it as $\log(b\ base\ a)$. From my experience, this notation has both advantages and drawbacks. When it comes to advantages:

• There is no possible confusion or misreading. It is especially not possible to confuse $\log(b\ base\ a)$ with $\log(a^b\ base\ ?)$, whereas it is indeed rather easy to misread $\log_a b$ as $\log a^b$. Generally speaking, I can't think of any possible misunderstanding of the notation itself, which was one of your biggest concerns.
• It is much easier to read out loud. As a direct consequence of the "no misunderstanding", there is no possibility for an out loud pronunciation of $\log(b\ base\ a)$ apart from "the logarithm of b to base a".
• It is impossible to mistake log as an independent mathematical object. You do not raise this particular issue in your question; however it does appear in brendansullivan's answer, as well as in Benjamin Dickman's link (student's misconceptions about the logarithm). For example, it is impossible to cancel the $\log$ in $\frac{\log(100\ base\ 2)}{\log(10\ base\ 2)}$, since $\frac{(100\ base\ 5)}{(10\ base\ 5)}$ cannot make sense.

When it comes to drawbacks:

• It is an unofficial notation. Therefore, it is highly likely that the students will have to learn the correct notation when the time comes. However, this should stand no real difficulty, given that when you have understood what you're talking about, the notation isn't that much of a big deal. They may even find the correct notation shorter and more pragmatic.
• It is rather long and not pragmatic. The usual notation has been designed to meet the needs of higher levels practice. It is therefore difficult to find a more efficient one. However, the usual exercises you practice on when you're learning about logarithm don't require a page-long proof or discussion, and a bit longer notation doesn't hurt that much.

To put it in a nutshell, considering the drawbacks and advantages, my advice would be to force yourself and your students into using the $\log(b\ base\ a)$ notation at first, to slowly introduce the topic and the use of logarithm. Once you've made sure that all of your students do understand what is going on and how the logarithm is used, including the formula you've presented, you can tell them about the usual notation. My opinion is that they will adapt very quickly to the real notation, and that they'll have overall less trouble in learning about the logarithm's properties.

Although it might not directly help you, given that the textbook you're using is already written in a different way, I hope this will be useful when it comes to your student's understanding of the logarithm.

Answer 3

The more general issue is that your students may simply be unfamiliar with reading and writing subscripts (irrespective of their particular use for logarithms). For example, I know that in my statistics classes the notation $z_{\alpha / 2}$ will frequently be transcribed incorrectly as $z \alpha /2$... or related manglings, like if we want $z_{\alpha / 2}$, and $\alpha = 0.10$, then $z = 0.05$.

I suppose if you had all the time in the world, then you could have a writing exercise where everyone in the class writes down three or four copies of a given (subscript) expression, and checks with a student next to them to see if they've written it correctly. Personally, I find that I never have time for that level of in-class inspection, even though I've attempted it/written it in my lecture notes in the past.

So my recommendation is that you've got to commit to getting into these details when grading and take points off if this syntax is incorrect (even if the final answer is otherwise correct). That's the only way that you'll draw a student's attention to fix the issue; and otherwise they'll be permanently confused by what they're reading and writing. I'm pretty sure that I could write large and talk until I'm blue in the face to be careful of this, and some students would still be blind to the subtlety of the notation until they see points coming off the test from it.

Answer 4

I frequently repeat the line "it's all about the base," which gets a chuckle, but then I say the goal of logs is to solve for the power.

A log is a power.

The power that gets us from that base to the result, so the second, false, equation would obviously be wrong.