Hi all, I wonder what is the best way to introduce logarithms when you're teaching. My "approach #1" is the one I consider the most natural: You introduce exponential functions as f(x) = bx, and ask what is the derivative. It turns out df/dx = lim(h->0) (bh-1)/h bx. Now, actually ln(b) = lim(h->0) (bh-1)/h. You could give this to the students as a definition of ln(). Now you define e by ln(e) = 1 and so on... This is unsatisfactory since ln() should be defined as the inverse of exp(). And since it doesn't give a recipe to compute e. My "approach #2" is that you don't tell the students anything about logarithms until you have defined e. Works like this: We want f'(x) = f(x), so we look for a number e which satisfies lim(h->0) (eh-1)/h = 1. Let's say h = 1/n, so lim(n->[oo]) n(e1/n-1) = 1, or e = lim (n->[oo]) (1 + 1/n) Then you introduce exp(x) = ex, and, as it's inverse, ln(x). Now, let f(x) = bx = exp(x ln(b)), we find df/dx = ln(b) * f(x), meaning the limes appearing above equals ln(b). This is also unsatisfactory since only the natural logarithm is introduced, not any other logarithms. So, my "approach #3" is, you introduce all the logarithms (i.e. logb() as inverse of b()), and then introduce e (as in approach #2), and then ln() as inverse of exp(). This is also unsatisfactory, since you don't use d/dx bx = ln(b)* bx. Which approach do you think is the best?