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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) = b^{x}, and ask what is the derivative.

It turns out

df/dx = lim(h->0) (b^{h}-1)/h b^{x}.

Now, actually

ln(b) = lim(h->0) (b^{h}-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) (e^{h}-1)/h = 1.

Let's say h = 1/n, so

lim(n->[oo]) n(e^{1/n}-1) = 1,

or

e = lim (n->[oo]) (1 + 1/n)

Then you introduce exp(x) = e^{x}, and, as it's inverse, ln(x).

Now, let

f(x) = b^{x}= 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. log_{b}() 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 b^{x}= ln(b)* b^{x}.

Which approach do you think is the best?

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# Introducing logarithms?

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