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Topic: definitions of exp/log
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John Hagood

Posts: 8
Registered: 12/6/04
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),

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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