Section 4.3: Logarithmic Functions
Instructor: Ms. Hoa Nguyen
([email protected])
1
Logarithmic Functions
An
exponential function
y
=
f
(
x
) =
a
x
with
a >
0,
a
6
= 1 is
onetoone
. Then its inverse
function
f

1
is the
logarithmic function
x
=
f

1
(
y
) =
log
a
y
.
In other words,
L
=
R
p
⇐⇒
p
=
log
R
L
.
Consequently, we have:
•
log
a
1 = 0 because
a
0
= 1.
•
log
a
a
= 1 because
a
1
=
a
.
Example 1
:
Example 2
:
1
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1
(
y
) =
log
a
y
for all
y
in the range of
f
(
x
) =
a
x
. The same as saying:
f

1
(
x
) =
log
a
x
for all
x
in the range of
f
(
x
) =
a
x
.
y
=
log
a
x
is read as “
y
is the logarithm to the base
a
of
x
”.
Since the exponential function and the logarithm function are inverses, denote
f
(
x
) =
log
a
x
and
f

1
(
x
) =
a
x
. From Section 4
.
2, we know that:
•
The domain of the exponential function
f

1
is the set of ALL real numbers.
•
The range of the exponential function
f

1
is the set of positive real numbers.
Then, what can we say about the domain and range of
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 Spring '11
 Nuegyen
 Exponential Function, Derivative, Logarithmic Functions, Inverse function, Logarithm, Ms. Hoa Nguyen

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