1549901369-Elements_of_Real_Analysis__Denlinger_

426 Chapter 7 11 The Riemann Integral

For those who skipped Section 5.6. The only previous definition we
must be consistent with is er, for rational numbers r. By Remark 7.7.8 (b),
exp(r) =er whenever r is a rational number. We extend this to all real numbers
x by defining

Vx E JR, ex= exp(x).

For those who studied Section 5.6. As we have noted above, for every
rational number r, er= exp(r). Thus, ex and exp(x) are continuous everywhere
on JR and agree on the dense set Q. Therefore, by Exercise 5.1.29,

Vx E JR, ex = exp(x).

Regardless of whether or not we studied Section 5.6, we see that the func-
tions ex and exp(x) are identical. Accordingly, we shall no longer use the no-
tation exp(x); we shall use ex exclusively.

The so-called "laws of exponents" described in the next theorem, were
proved in Section 5.6. It is interesting, especially for those who skipped that
section, to find that these laws can be derived directly from the definition of
the exponential function as the inverse of the logarithm function. We do so in
the next theorem.

Theorem 7.7.10 (Laws of Exponents) Vx, y E JR,

(a) exey = ex+y

(c) e^0 = 1

(e) ln(ex) = x

(b) ex /eY = ex-y

(d) e - x = l/ex

(f) elnx = X

Proof of (a): Let u =ex and v = eY. Then lnu = x and lnv = y. Thus,
by the laws of logarithms (Theorem 7.7.3),

x + y = ln u + ln v = ln(uv) = ln(exeY).

That is, by Definition 7.7.7, ee+y = eXeY. •

Theorem 7. 7 .11 (Derivative of e"') The function ex is differentiable ev-
d
erywhere, and dx ex = ex.

Proof. See the proof of Theorems 7.7.2 (d) and 6.2.9 (b). •

Theorem 7. 7.12 ( ekx and the Differential Equation f' = kf) Let k ER
If f :JR ---> (0, oo) satisfies the differential equation f' = kf everywhere on JR,
then 3c E JR 3 Vx E JR, f(x) = cekx_ Moreover, c = f(O).