5.6 *Exponentials, Powers, and Logarithms 277show that loga x = ln x / ln a. In a later chapter of elementary calculus, power
series expansions of these functions are easily derived from these definitions.
A serious disadvantage of this approach is that it requires us to delay defining
these functions until the theory of Riemann integration has been developed,
later in the course.
Another approach often taken in real analysis textbooks is to delay the
actual definition of ex until power series have been studied. Indeed, we take
this approach in our Chapter 8. In a straightforward manner in Section 8.8
oo n
we define ex = l:= ;. Then, using the machinery of power series, we prove
n=O n.
that this function has all the algebraic properties expected of the exponential
function. We obtain its derivative as a power series, and easily see that it is
ex itself. We then obtain its inverse function, ln x, and show it to have the
expected algebraic properties and derivative.
The power series approach provides quick and direct definitions, even though
it leaves significant algebraic details to be worked out using power series meth-
ods. One advantage of this approach is that it makes obvious a beautiful con-
nection between the exponential function and the sine and cosine functions.
When complex numbers are allowed, it leads quickly to the elegant formula
eie = cos B + i sin B, from which Euler's famous formula, ein = -1, is an im-
mediate consequence. A big disadvantage of this approach, however, is that it
requires us to postpone defining the exponential and logarithm functions until
after the theory of power series has been developed.
The approach we take here has an elegance of its own. While foregoing the
sophisticated power of either the Riemann integral or power series, it demon-
strates the power of the concepts already developed in this course, particularly
the ideas of supremum and infimum, limits of sequences and functions, conti-
nuity, monotonicity, and even the denseness of the rational numbers in R
The problem of defining ax, where a > 0 and xis an arbitrary real number,
is more subtle than one might expect. In the exercises at the end of Section
5.5, we indicated how we can use the inverse function theorem for continuous
functions to define a^1 1n (n E N) and consequently ar for all rational numbers
r. It is straightforward to show that the function f ( r) = ar, defined in this
manner for rational numbers r , has all the expected algebraic properties. [See
Exercise 5.5.15.] In the present section we take up the problem of extending
the domain of this function in a natural way to the set of all real numbers.
A related function is g(x) = xe, where c is a constant real number. In
elementary calculus, we usually presume the existence of this function, and
we intuitively accept the claim that its derivative is g' ( x) = cxe-l. Without
a rigorous definition of this function, however, we cannot even prove that it
is continuous, let alone differentiable. In this section we place this function on
a firm foundation. It turns out that we use this function g(x) = xe (constant