4
EXAMPLE 3
416
Chapter 3
Exponential and Logarithmic Functions
Figure 3.18
shows the graphs of
and
in
by
viewing rectangles. Are
and
the same? The graphs
illustrate that
and
have different domains. The graphs are only the same if
Thus, we should write
ln
x
2
=
2 ln
x
for
x
7
0.
x
7
0.
y
=
2 ln
x
y
=
ln
x
2
2 ln
x
ln
x
2
3

5, 5, 1
4
3

5, 5, 1
4
y
=
2 ln
x
y
=
ln
x
2
Properties for Expanding Logarithmic Expressions
For
and
1.
Product rule
2.
Quotient rule
3.
Power rule
log
b
M
p
=
p
log
b
M
log
b
a
M
N
b
=
log
b
M

log
b
N
log
b
1
MN
2
=
log
b
M
+
log
b
N
N
7
0:
M
7
0
Expand logarithmic expressions.
Section 3.3
Properties of Logarithms
417
Try to avoid the following errors:
Incorrect!
log
b
1
MN
p
2
=
p
log
b
1
MN
2
log
b
M
log
b
N
=
log
b
M

log
b
N
log
b
a
M
N
b
=
log
b
M
log
b
N
log
b
1
M
#
N
2
=
log
b
M
#
log
b
N
log
b
1
M

N
2
=
log
b
M

log
b
N
log
b
1
M
+
N
2
=
log
b
M
+
log
b
N
Study Tip
The graphs show that
In general,
y
2
=
ln
x
+
ln 3
y
1
=
ln (
x
+
3)
[
−
4, 5, 1] by [
−
3, 3, 1]
log
b
1
M
+
N
2
Z
log
b
M
+
log
b
N
.
ln
(x+3)
ln
x+
ln
3.
y
=
ln
x
shifted
3 units left
y
=
ln
x
shifted
ln
3 units up
Expanding Logarithmic Expressions
Use logarithmic properties to expand each expression as much as possible:
a.
b.
Solution
We will have to use two or more of the properties for expanding
logarithms in each part of this example.
a.
Use exponential notation.
Use the product rule.
Use the power rule.
b.
Use exponential notation.
Use the quotient rule.
Use the product rule on
Use the power rule.
Apply the distributive property.
because 2 is the
power to which we must raise
6 to get 36.
Check Point
4
Use logarithmic properties to expand each expression as much
as possible:
a.
b.
log
5
¢
1
x
25
y
3
≤
.
log
b
A
x
4
1
3
y
B
1
6
2
=
36
2
log
6
36
=
2
=
1
3
log
6
x

2

4 log
6
y
=
1
3
log
6
x

log
6
36

4 log
6
y
=
1
3
log
6
x

1
log
6
36
+
4 log
6
y
2
log
6
1
36
y
4
2
.
=
log
6
x
1
3

1
log
6
36
+
log
6
y
4
2
=
log
6
x
1
3

log
6
1
36
y
4
2
log
6
¢
1
3
x
36
y
4
≤
=
log
6
¢
x
1
3
36
y
4
≤
=
2 log
b
x
+
1
2
log
b
y
=
log
b
x
2
+
log
b
y
1
2
log
b
1
x
2
1
y
2
=
log
b
A
x
2
y
1
2
B
log
6
¢
1
3
x
36
y
4
≤
.
log
b
1
x
2
1
y
2
EXAMPLE 4
Condensing Logarithmic Expressions
To
condense a logarithmic expression
, we write the sum or difference of two or
more logarithmic expressions as a single logarithmic expression. We use the
properties of logarithms to do so.
Study Tip
These properties are the same as
those in the box on page 416. The
only difference is that we’ve reversed
the sides in each property from the
previous box.
Properties for Condensing Logarithmic Expressions
For
and
1.
Product rule
2.
Quotient rule
3.
Power rule
p
log
b
M
=
log
b
M
p
log
b
M

log
b
N
=
log
b
a
M
N
b
log
b
M
+
log
b
N
=
log
b
1
MN
2
N
7
0:
M
7
0
Condensing Logarithmic Expressions
Write as a single logarithm:
a.
b.
Solution
a.
Use the product rule.
We now have a single logarithm.
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 Fall '16
 Kirkpatrick
 Physics, Derivative, pH, Natural logarithm, Logarithm, natural logarithms