Calculus: Analytic Geometry and Calculus, with Vectors

9.1 Exponentials and logarithms 481

physical theory of hydrogen, is neither so brief nor unimportant that it
is unworthy of several hours of study.

We start here with a given number a, the base of our exponential func-
tion ax, for which a > 1. We suppose that ax has been defined for rational

numbers x and that, if x and y are rational numbers for whichx = P/Q,

then

(9.11) axav = ax+l, (ax)/ = axv, ax = (aP)uQ = (a1'Q)P.

For the purpose of learning about mathematics in general and about
the exponential function ax in particular, we
make a direct frontal attack upon the problem
of learning more about ax when x is rational. I y
We then use this information as a basis for def- y=ax
inition and study of ax when x is real. Our xrational 0
first step is to calculate ax for several values of
x and to start making a graph of y = ax as in 0
Figure 9.12. We mark only the points whose 0
coordinates we have calculated and overcome
the primitive urge to draw a "smooth curve" e
through these points. We have experimental o
evidence that 0 < ax < ax+h when x and h are 0
rational and h > 0, and this matter should^0

00001

be investigated. When x = P/Q, a1"Q is the -2 -1^012 x

positive number r for which rQ = a and Figure 9.12

ax = (a1"Q)P = rP > 0, so ax > 0. It turns out

that we can obtain a huge amount of information from the identity

(9.13) ax+h - ax = (a'' - 1)az.

For a modest beginning we suppose that h = p/q, where p and q are
positive integers, and show that ah > 1. When f(x) = x4, we see that
f(1) = 1 and that f(x) is continuous and increasing when x > 0. Since
a > 1, the one and only number x1 for which f(xi) = a must exceed 1.
Thus xl = a and a119 = x1, where x1 > 1, and hence ah = xi > 1. Thus,
when h > 0, the right member of (9.13) is the product of two positive
numbers and hence is positive. This implies the following theorem.
Theorem 9.14 If x and h are rational and h > 0, then

(9.141) 0 < ax < ax+h.

This means that, as a function of the "rational variable" x, the function

having values ax is positive and increasing. Proof of this fundamental

fact represents a triumph of mind over matter, but one triumph is not

enough and we must continue. Tables and slide rules and brute-force

calculations with pencils produce experimental evidence in support of the

following theorem.