(ii)
The range of
exp
a
is the set of positive real numbers.
(iii)
The function
exp
a
is continuous.
(iv)
The function
exp
a
converts addition into multiplication, i.e.
exp
a
(
x
+
y
) = exp
a
(
x
)
·
exp
a
(
y
)
.
(v)
The value of
exp
a
(
x
)
when
x
= 1
is
exp
a
(1) =
a.
7
My favorite is Stewart:
Calculus: Early Transcendentals
.
53
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Chapter 0 / Exercise 3
Mathematical Applications for the Management, Life, and Social Sciences
Harshbarger
Expert Verified
§
16.2.
The more familiar notation for the exponential function is
exp
a
(
x
) =
a
x
so parts (iv) and (v) of Theorem 16.1 take the more familiar form
a
x
+
y
=
a
x
·
a
y
,
a
1
=
a.
(
♥
)
This implies that, for integers, exponentiation is repeated multiplication,
8
e.g.
a
3
=
a
1+1+1
=
a
1
·
a
1
·
a
1
=
a
·
a
·
a
. Using (
♥
) repeatedly gives
a
nx
=
a
x
+
x
+
· · ·
+
x
n
=
a
x
·
a
x
· · · ·
a
x
n
= (
a
x
)
n
so taking
x
= 1
/n
and using (
♥
) again proves that for any positive integer
n
a
1
n
=
n
√
a
the
n
th root of
a
(i.e. the functions
b
=
a
n
and
a
=
b
1
n
are inverse functions).
It follows easily that parts (iv) and (v) of Theorem 16.1 determines the value
a
x
uniquely when
x
is a rational number
9
and, as every real number is a limit
of rational numbers, this shows that the conditions of Theorem 16.1 uniquely
determine the exponential function. In Section 38 we give a formula for exp
a
and show that it satisfies the conditions of Theorem 16.1.
§
16.3.
The following familiar laws of algebra all follow easily from equation (
♥
).
a
0
= 1
,
a
x
+
y
=
a
x
·
a
y
,
a

x
=
1
a
x
,
a
1
=
a,
(
ab
)
x
=
a
x
·
b
x
,
(
a
p
)
x
=
a
p
·
x
.
For example, the reason why (
ab
)
x
=
a
x
·
b
x
is that both sides satisfy the
conditions of Theorem 16.1 (reading
ab
for
a
). Similarly, to prove (
a
p
)
x
=
a
p
·
x
note that both sides (as functions of
x
and reading
a
p
for
a
) satisfy the conditions
of Theorem 16.1 (at least if
p
= 0; if
p
= 0 both sides equal 1).
8
Just as multiplication is repeated addition.
9
A
rational number
is a ratio of integers, i.e. a fraction.
54
Theorem 16.4.
For
a >
1
, the exponential function is differentiable, strictly
increasing, and satisfies so
lim
x
→∞
a
x
=
∞
,
and
lim
x
→∞
a
x
= 0
.
These will be proved in section 38. That the exponential function
y
=
a
x
is
strictly increasing means that
a
x
1
< a
x
2
for
x
1
< x
2
. (For example, 2
3
<
2
4
.)
This, combined with the Intermediate Value Theorem 9.7 means that the range
of the exponential function is the set of all positive numbers and the graph of
the exponential function passes the vertical line test so the exponential function
has an inverse function.
Definition 16.5.
The inverse function to the exponential function
exp
a
(
x
) =
a
x
is called the
logarithm function base
a
:
y
=
a
x
⇐⇒
x
= log
a
(
y
)
.
The range of log
a
is all real numbers, the domain is all positive rea
numbers, and
log
a
(
a
x
) =
x,
a
log
a
(
y
)
=
y
by the Cancellation Law for inverse functions.
Remark 16.6.
Since (1
/a
)
x
= 1
/a
x
=
a

x
, similar statements hold if
a <
1.
Thus for
b <
1
lim
x
→∞
b
x
= 0
and
lim
x
→∞
b

x
=
∞
and the function
b
x
is strictly decreasing.