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definitions of exp/log
Posted:
Oct 18, 2001 2:30 PM
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Here is another way to definite exp(x), appropriate for first year calculus
or early in advanced calculus (prior to developing the derivative).
(1) Start by extending a^x from the rationals to a continuous function on
the reals.
(2) Consider the functions x/a^x for constant a>1. In first year calculus,
graphical evidence suggests that each such function has exactly one
maximum, and that location of the maximum decreases as x increases. Take e
to be the value of a such that the maximum occurs at x = 1.
(3) It is now an easy and entertaining exercise to obtain quite rigorously
the derivative of e^x. Furthermore, the proof of the traditional limit
lim_{n- > infinity}(1+1/n)^n = e is pleasantly quick as well.
This will appear in the November 2001 issue of College Mathematics Journal.
For a more advance class - i.e., Introduction to Analysis or Advance
Calculus - the details of (2) can be filled in as follows (these are
reasonable exercises for students)
(2a) For each a>1, x -> x/a^x attains a maximum. The extreme value theorem
will get this, but if that is not yet in hand, it is an easy exercise to
show that for each a>1 here is a cut (L,R) of the rationals such that x/a^x
increases strictly on L and decreases strictly on R . This approach also
shows that the location of the maximum is unique.
(2b) The maximum of x/a^{kx} occurs at x_a/k where x_a is the location of
the maximum of x/a^x.
(2c) There is a unique number e such that x/e^x has a maximum at x =
1. (The uniqueness can be proved using (2b) and the fact that for a given
a>1, x->a^x maps onto the positive reals. The intermediate value theorem
makes this immediate, but it is not needed in light of elementary
properties of a^x over the rationals (x->a^x is strictly increasing and
unbounded above, a^t-a^s <= a^{t+1}(t-s) for s<t, and the extension in (1)
) if one prefers to avoid it.
An alternative approach not mentioned already in this thread appears in
(John Kemeny, The exponential function, Amer. Math. Monthly, 64 (1957),
158-160).
John W. Hagood
Department of Mathematics and Statistics
PO Box 5717
Northern Arizona University
Flagstaff, AZ 86011-5717
Phone: 520-523-6879
Fax: 520-523-5847
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