This preview shows pages 1–2. Sign up to view the full content.
MATH 191, Sections 1 and 3
Calculus I
Fall 2007
Section 1.5: Fun with Exponential Functions
In order to understand the
function
f
(
x
) =
a
x
(here, and
always,
a >
0), we can play around with what
a
x
must be for various values of
x
:
1.
Positive integers.
Of course,
a
1
=
a
,
a
2
=
a
·
a
, and so forth. That is,
a
n
is
a
multiplied by itself
n
times, whenever
n
is any positive integer.
This gives our ﬁrst rule for exponents, as follows: since
a
m
·
a
n
= (
a
·
a
· ··· ·
a
)(
a
·
a
· ··· ·
a
)
where there are
a
s in the ﬁrst term, and
a
s in the second, we have
a
m
·
a
n
=
for any positive integers
m
and
n
.
2.
Other integers.
For the above rule to hold for
all
integers, not just
positive ones, we must be able to let
n
= 0:
a
m
·
a
0
=
a
m
+0
=
a
m
.
For this to be so, we must have
a
0
=
for any
a
.
Also, letting
m
=

n
, we get
a

n
·
a
n
=
a

n
+
n
=
a
0
= 1
.
From this it follows that
a

n
=
must be true, ﬁnishing our compu
tation of
a
n
for any integer. Note this also gives another, related, rule for
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.