1549901369-Elements_of_Real_Analysis__Denlinger_

7.7 *Elementary 'Ifanscendental Functions 425

(d) The range ofln: (0, +oo) --t JR is JR; the graph is as shown in Figure 7.11.

y

2

(e2,2)

x

Figure 7.11

THE EXPONENTIAL FUNCTION

Definition 7.7.7 (The Function exp(x)): Since the function ln: (0 , +oo) --t
JR is continuous, strictly increasing, 1-1, and onto, Corollary 5.5.3 assures us that
it has an inverse ln-^1 : JR --t (0, +oo) which is continuous, strictly increasing,
1-1, and onto. We denote that inverse (temporarily) by

exp(x) = ln-^1 x.

That is , y = exp(x) if and only if x = lny.

Remarks 7.7.8

(a) expx > 1, ifx > 0.

expx= 1, ifx=O.

0 < exp x < 1, if x < 0.

(b) Vr E Q,exp(r) =er.

(c) For any sequence {rn} of rational numbers converging to x,

exp(x) = lim exp(rn) = lim ern.

n-.oo n-too

Definition 7.7.9 (The Function e"')

We now come to the problem of defining ex, for arbitrary real numbers x
(in particular, for irrational numbers x) in such a way that our definition is
consistent with previously agreed-upon definitions.