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Supplemental Material for Section 7.4:
Exponential Functions
In this section the general exponential function,
a
x
is deﬁned and some of
its properties are presented. This document contains material that should have
been included in the text. Recall that
a
x
has already been deﬁned for many
values of
x
. For example
a
1
2
denotes
√
a.
In fact
a
r
has been deﬁned for any
rational number
r
=
m
n
by
a
r
= (
n
√
a
)
m
. Because only positive numbers have
n
th
roots, it’s natural to assume that
a >
0 which we do. To see how to use the
function
e
x
to deﬁne
a
x
, note that for
r
a rational number,
a
r
=
e
ln
a
r
=
e
r
ln
a
.
The deﬁnition of
a
x
is accomplished by simply insisting that the same formula
holds for any real number
x
; not just rational numbers.
Deﬁnition.
Let
a >
0
and let
x
be any real number. Then
a
x
=
e
x
ln
a
.
The basic laws of exponents will now be established from this deﬁnition using
the properties of the function
e
x
and the laws of logarithms. We ﬁrst note that
for
a >
0 and any real number

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