Calculus: Analytic Geometry and Calculus, with Vectors

9.1 Exponentials and logarithms 483

very substantial,and we proceed to use it to define and study ax when x
is real. Let x be a real number which is not necessarily rational. In
case x is rational,the fact that ar is an increasing function of the rational
variable r implies that

(9.17) ax = l.u.b. ar,

r5z

where the right side is the least upper bound of the set of numbers ar

for which r is rational and r 5 x. In case x is irrational, we define the

number ax by this same formula (9.17). To use this definition, we need

only a very little information about least upper bounds of sets of num-

bers. We need only the elementary fact that if each number in a set S,

is also a number in a set S2, then the least upper bound of the first set

must be less than or equal to the least upper bound of the second set.

Suppose xi, and x2 are given real numbers for which xl < x2 and we choose

two rational numbers rl and r2 such that

xl <Ii<r2 <X2.

Then, since art < ar2,

ax'=1.u.b.ar<1.u.b.ar=ar'<all =l.u.b.ar<_l.u.b.ar=all,

r5zi r5ri r<rx r5z2

and this proves that ax is an increasing function of x. If 0 < & < M,

we can choose a positive integer n such that an > M and a-, < &.

This and the fact that ax is positive and increasing implythat az -- 0

as x -+ - w and ax --* oo as x -> c*. To prove that ax is continuous

(and in fact uniformly continuous) over the interval x <M, we can

employ Theorem 9.16. Let e > 0. We can then choose a & > 0 such

that jar,- ar'I < e whenever ri < M, r2 < M, and Ir2 - ril < &.

Whenever Xi and x2 are real numbers for which xi <M, X2 < M, and

IX2 - x11 < &, we can sandwich x, and x2 between rational numbers

ri and r2 for which rl < M, 72 < M, and jr2 - ril < &. The fact that

ax is increasing then implies thatlax,- ax'I < jar' -ar2l < E.

To complete our study of the basic theory of ax when xis real, we must

prove the formulas

(9.18) axav = ax-w, (ax)y = axv.

Since ax is continuous, we can let rl, r2,. and si, s2,. be sequences

of rational numbers converging to x and y so that

axav = (lim a*^)(lim a"") = lim

n-t n- m

= lim a,-+O- = ax*"

fl-.+02

and the first formula is proved. To prove the second formula, we vary