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Section 4.4. Properties of Logarithms
Properties of Logarithms:
•
log
a
1=0
•
log
a
a
=1
•
log
a
M
r
=
r
log
a
M
•
log
a
a
r
=
r
•
a
log
a
x
=
x
•
log
a
M
=log
a
N
implies
M
=
N
•
log
a
(
MN
)=log
a
M
+log
a
N
•
log
a
(
M
N
a
M

log
a
N
The following examples illustrate the above rules.
Example.
log (
√
x
(
x
+1)
2
)
√
x
+log(
x
2
=
1
2
log
x
+2log(
x
Example.
log
√
x
(
x
+1)
2
√
x

log (
x
2
=
1
2
log
x

2log(
x
Example.
log 3
2

3log5
=log3
2

log 5
3
3
2
5
3
9
125
Example.
Suppose
y
=3
x
2
,then
ln
y
=ln3
x
2
=ln
x
2
+ln3
=2ln
x
Interesting observation: In the previous example,
y
was a power function of
x
(i.e.
y
x
2
), and
this implied that ln
y
was a linear function of ln
x
, i.e.,
y
x
2
implied that ln
y
x
+ln3. The
example illustrates the following general rule:
1
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View Full Documenty
=
ax
b
if and only if ln
y
=
b
ln
x
+ln
a
The above implies that if a plot of ln
x
against ln
y
, for some set of data, is a straight line, then one
would expect
y
to be a power function of
x
for that set of data.
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 Fall '11
 Kutter
 Algebra

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