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definitions of exp/log
Posted:
Oct 18, 2001 2:30 PM


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^ta^s <= a^{t+1}(ts) 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), 158160).
John W. Hagood Department of Mathematics and Statistics PO Box 5717 Northern Arizona University Flagstaff, AZ 860115717 Phone: 5205236879 Fax: 5205235847

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