Calculus: Analytic Geometry and Calculus, with Vectors

9.2 Derivatives and integrals of exponentials and logarithms 493

Since a0 has a positive derivative, Theorem 8.33 implies that its inverse
log,, x, defined over the infinite interval x > 0, is differentiable. Hence
we can differentiate the first of the formulas

(9.251) alog. x = x, Aa'°gax z log.x = 1

with the aid of the chain rule to obtain the second and hence obtain the
second formula in (9.25). Conversely, if the second formula in (9.25)

is known to hold, then differentiation of the members of the formula
log. ax = x gives the first formula in (9.25). Since logo 1 = 0, replacing
x by t in the second of the equivalent formulas (9.25) and integrating
over the interval from 1 to x gives the formula

(9.252) Alog.x=flo 1dt.

tt

Since loga a = 1, putting x = a gives the formula

(9.253) A=

f

(aIdt

which clarifies the relation between a and A.

Exponential functions and their derivatives and integrals occur so

often that the constant A in the above formulas would, if allowed to

survive, be an insufferable nuisance. Hence we can relish the proof that

we can choose a to make A = 1. If we let f(x) denote the right member

of (9.252), then f is continuous and increasing over the whole interval

x>0,f(1)=0,and

r4

(9.254) f (4) =J tdt > + +

4

> 1.

1 t

There is therefore exactly one positive number e for which 1 < e < 4

and the constant A in (9.253) and preceding formulas is I when a = e.

Thus all of the above formulas are correct when a = e and A = 1.

In particular, we have proofs of the basic formulas

(9.26)

d

ex - ex, dx logx =

1

Y

and hence also of their companions

(9.261)

J

ex dx = ex + c, fidx=logx+X

Setting a = e and A = 1 in (9.24) and (9.253) gives the formulas

(9.262) exdx=e-1,

fo fi

1dx=1,

X