I need to know more (as always, right?) I need to start the conversation with a simpler logarithmic equation. How would this student think about ln(x)=12? Will they divide by e here too? If so, then we’ve got to go back to first principles. If not, then I want to ask the student why they divided by e in one instance but not the other; elicit from them what the similarities and differences are in the two situations.

There are a lot of things I like about this response, but one of them is that it draws out a teaching move that often works: bringing the conversation back to a simpler case. As I think about the possibilities for this site moving forward I tend to think about amassing a huge database of student errors. But just as important, we’re amassing a huge database of teacher responses. If we could analyze these responses into a few techniques or principles, I think that would be a nice step forward.

This is a boring answer, but I would think this student was simply clueless regarding how to do this problem, grabbed an old process he knew and applied it.

I don’t think this answer is boring at all! For one thing, I love to see applications of “relate this to a problem I have solved before,” even if the execution is imperfect.I also like getting to see that the student clearly sees a relationship between ln and e (win!) and has some good ideas about using inverse operations to make unknown values apparent.Looks like the student wasn’t using a calculator… it’s pretty clear they didn’t type ln(4) / e into a calculator, so I’m not convinced they were definitely thinking of dividing, more of undoing. If they had the expression ln(4*6x) = ln 48, it would have been appropriate to “undo” the ln operation and find values of x such that 4 * 6x = 48.I’m curious to hear the student’s answer to, “what makes this problem hard?” Does the student realize on some level that this problem is different from ln(24x) = ln 48?Is the bigger problem here with the concept order of operations and undoing two separate applications of ln in one swoop, or is it the concept thinking that ln is multiplicative, or is it failing to apply the method/strategy of “turn this problem (with two ln terms) into one I have solved before (with one ln term)?”

You never *really* know, but I would suspect that the student latched onto the idea that “ln and e undo each other, so I just have to use e to get the equation to be nice.” However, when it came to actually using e, the student didn’t quite know what to do (it’s unclear if the student thought what s/he was doing was right or if the student did this in a moment of desperate panic… I’d bet on the former). So I’d say that the student had some conception of the idea that “ln and e are inverse operations” but still doesn’t understand the weird notation for ln. This becomes clear because not only does it look like the student thinks that s/he can divide by e to “undo” the ln function, but also because s/he doesn’t immediately recognize that ln(4)+ln(6x) can be simplified. Honestly, I’d bet the student was so used to solving problems like: ln(4)=ln(5x) and just skipped to 4=5x (by, in their minds, dividing out the lns) that s/he just in their minds generalized that procedure to all problems of the form. I typed this before looking at the comments above (to see if I’d come up with something different), but it seems Max and I are on the same page!

This is one step beyond a common mistake. As Sameer said, they latch on to the fact that e and ln are inverse operations. You have to explicitly go through the reason that applying a base to each term is not “doing the same thing to both sides” – start with a 2 + 3 = 5 and apply a base of 2 or 10 to each term. I have not seen the division bar before, but I very much doubt that they intend it to mean division. Some teachers draw a division bar through the equal sign like this and then use the word “cancel” when they are dividing out fators on each side. I suspect the student was thinking “cancel”.

It seems like the student is trying to “exponentiate” the equation in order to cancel the logs, but didn’t realize that you can only exponentiate once on each side of the equals sign, just like if you are adding 3, you have to add it once to each side only. I have seen my students make this same mistake because they forget the properties of condensing logs so don’t remember what to do.I’m not sure what the student meant by drawing the line, if he really meant division, or if he actually had the proper idea, just didn’t know how to execute it.

As with many things, kids know they need an inverse operation to “undo” something, or make it simpler, or move back a step. I teach younger kids than (I think) you do, but my students constantly confuse multiplication and exponentiation. (For instance, many of mine, early in the year, will tell me that 7^3 is 21.) This looks like that, and also that the student isn’t applying the rules of logs.I agree with mr bombastic that they’ve latched on to the fact that e and ln are somehow inverses of each other, but they don’t know or remember anything more specific than that.

[…] week we posted a somewhat similar mistake involving logarithms. Here’s the student […]

Sam and Krystal, I think, are onto something here. I do not think that the student is dividing by e. Instead they are adding a base of e to each side and, mistakenly, concluding that is x + y = z, then e^x + e^y = e^z
This is a classic, slightly more sophisticated, version of sqrt(x^2 + y^2) = x + y

I saw this referred to once as the Student Universal Distributive Law

## 10 replies on “Dividing by “e””

I need to know more (as always, right?) I need to start the conversation with a simpler logarithmic equation. How would this student think about ln(x)=12? Will they divide by e here too? If so, then we’ve got to go back to first principles. If not, then I want to ask the student why they divided by e in one instance but not the other; elicit from them what the similarities and differences are in the two situations.

There are a lot of things I like about this response, but one of them is that it draws out a teaching move that often works: bringing the conversation back to a simpler case. As I think about the possibilities for this site moving forward I tend to think about amassing a huge database of student errors. But just as important, we’re amassing a huge database of teacher responses. If we could analyze these responses into a few techniques or principles, I think that would be a nice step forward.

This is a boring answer, but I would think this student was simply clueless regarding how to do this problem, grabbed an old process he knew and applied it.

I don’t think this answer is boring at all! For one thing, I love to see applications of “relate this to a problem I have solved before,” even if the execution is imperfect.I also like getting to see that the student clearly sees a relationship between ln and e (win!) and has some good ideas about using inverse operations to make unknown values apparent.Looks like the student wasn’t using a calculator… it’s pretty clear they didn’t type ln(4) / e into a calculator, so I’m not convinced they were definitely thinking of dividing, more of undoing. If they had the expression ln(4*6x) = ln 48, it would have been appropriate to “undo” the ln operation and find values of x such that 4 * 6x = 48.I’m curious to hear the student’s answer to, “what makes this problem hard?” Does the student realize on some level that this problem is different from ln(24x) = ln 48?Is the bigger problem here with the concept order of operations and undoing two separate applications of ln in one swoop, or is it the concept thinking that ln is multiplicative, or is it failing to apply the method/strategy of “turn this problem (with two ln terms) into one I have solved before (with one ln term)?”

You never *really* know, but I would suspect that the student latched onto the idea that “ln and e undo each other, so I just have to use e to get the equation to be nice.” However, when it came to actually using e, the student didn’t quite know what to do (it’s unclear if the student thought what s/he was doing was right or if the student did this in a moment of desperate panic… I’d bet on the former). So I’d say that the student had some conception of the idea that “ln and e are inverse operations” but still doesn’t understand the weird notation for ln. This becomes clear because not only does it look like the student thinks that s/he can divide by e to “undo” the ln function, but also because s/he doesn’t immediately recognize that ln(4)+ln(6x) can be simplified. Honestly, I’d bet the student was so used to solving problems like: ln(4)=ln(5x) and just skipped to 4=5x (by, in their minds, dividing out the lns) that s/he just in their minds generalized that procedure to all problems of the form. I typed this before looking at the comments above (to see if I’d come up with something different), but it seems Max and I are on the same page!

This is one step beyond a common mistake. As Sameer said, they latch on to the fact that e and ln are inverse operations. You have to explicitly go through the reason that applying a base to each term is not “doing the same thing to both sides” – start with a 2 + 3 = 5 and apply a base of 2 or 10 to each term. I have not seen the division bar before, but I very much doubt that they intend it to mean division. Some teachers draw a division bar through the equal sign like this and then use the word “cancel” when they are dividing out fators on each side. I suspect the student was thinking “cancel”.

It seems like the student is trying to “exponentiate” the equation in order to cancel the logs, but didn’t realize that you can only exponentiate once on each side of the equals sign, just like if you are adding 3, you have to add it once to each side only. I have seen my students make this same mistake because they forget the properties of condensing logs so don’t remember what to do.I’m not sure what the student meant by drawing the line, if he really meant division, or if he actually had the proper idea, just didn’t know how to execute it.

As with many things, kids know they need an inverse operation to “undo” something, or make it simpler, or move back a step. I teach younger kids than (I think) you do, but my students constantly confuse multiplication and exponentiation. (For instance, many of mine, early in the year, will tell me that 7^3 is 21.) This looks like that, and also that the student isn’t applying the rules of logs.I agree with mr bombastic that they’ve latched on to the fact that e and ln are somehow inverses of each other, but they don’t know or remember anything more specific than that.

[…] week we posted a somewhat similar mistake involving logarithms. Here’s the student […]

Sam and Krystal, I think, are onto something here. I do not think that the student is dividing by e. Instead they are adding a base of e to each side and, mistakenly, concluding that is x + y = z, then e^x + e^y = e^z

This is a classic, slightly more sophisticated, version of sqrt(x^2 + y^2) = x + y

I saw this referred to once as the Student Universal Distributive Law