- #1
arcnets
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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?
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?
Last edited: