What you need to know is that the student work is the stuff typed in red, and that this came on a take-home quiz:

MathMistakes_e_1 MathMistakes_e_2


Why does this student think that ln(e) = x^1. Or did I get that wrong? Are there ideas that “feel similar” that he’s confusing, or is it something else?

(Thanks Taylor!)

  • What is frustrating to me looking at this is the fact that the student does not seem to recognize that e^x and a^x are structurally the same thing. It seems pretty clear that e is seen as an entirely different kind of thing. This carries over to the ln versus log base a question as well. It feels to me that this student is just trying to sort and memorize a whole bunch of facts rather than seeing any connections here. In the first question if the x^1 had been switched to 1^x we would have had a correct answer for an incorrect reason – these are always hard to navigate as a teacher.
    I think I would start by asking the student to look at a variety of a^x functions and emphasize the correct answer here, then bridge to something like pi^x before looking at the e^x. Try to convince the student of the stability of the pattern there. Perhaps correcting this would lead naturally to correcting the log questions.

    • Hao

      It is curious to me that the derivative of a^x is correct, but somehow incorrect for e^x. I’ve never remembered the general formula for d/dx(a^x), instead using d/dx(e^x) = e^x and then using a transformation and chain rule: a^x = e^(ln(a)*x).

  • It is possible that the student looked up answers for a^x and ln(a) but tried those with a=e on his/her own.

    • Laura C.

      I can definitely see this happening. Students have an unfortunate tendency to confuse x^1 and the number 1.

      Since ln (e) = 1, transferring from the general case to a = e would give you:

      e^x * ln(e) = e^x * 1 = e^x

      But just judging by the number of students to think that x^0 = x^1 instead of just 1, I’d say something similar is probably happening here. Instead of ln(e) = 1, the student has decided that ln(e) = x^1.

  • Perhaps a failed chain rule? That is to say, a student who knows d/dx e^{x^2} = (2x) e^{x^2}, d/dx e^{x^3} = (3x^2) e^{x^3} and extrapolates wrongly to d/dx e^x = (1 x) e^x (when it should be 1 x^0 e^x.)