Intermediate Algebra Unit 9: Logarithms Worksheet - Tangient Llc Page 9
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25Intermediate Algebra
Unit 9: Logarithms
Properties of Logarithmic Functions
If b, M, and N are positive real numbers, b ≠ 1, and p and x are real numbers, then:
Definition
Examples
=
0
written exponentially: b
= 1
log
1
0
b
=
1
written exponentially: b
= b
log
b
1
b
=
x
x
written exponentially: b
x
= b
log
b
x
b
=
=
log
x
,
log
7
b
x
10
7
b
where x > 0
10
=
+
log
9
x
log
9
log
x
3
3
3
=
+
log
MN
log
M
log
N
=
+
log
yz
log
y
log
z
b
b
b
1
1
1
5
5
5
2
=
−
log
log
2
log
5
4
4
4
M
5
=
−
log
log
M
log
N
b
b
b
N
7
=
−
log
log
7
log
x
8
8
8
x
=
x
log
6
x
log
6
=
2
2
p
log
M
p
log
M
b
b
=
4
log
y
4
log
y
5
5
−
=
+
log
3 (
x
) 4
log
5 (
x
) 2
=
6
6
M = N
log
M
log
N
if and only if
∴
−
=
+
b
b
3 (
x
) 4
5 (
x
) 2
Common Errors:
M
−
=
log
M
log
N
log
b
b
b
log
M
N
≠
−
b
log
M
log
N
b
b
log
M
log
N
cannot be simplified
b
b
log
N
b
+
=
log
M
log
N
log
MN
+
≠
+
b
b
b
log
(
M
) N
log
M
log
N
b
b
b
+
cannot be simplified
log
(
M
) N
b
=
p
p
log
M
log
M
b
b
≠
p
(log
M
)
p
log
M
b
b
cannot be simplified
p
(log
M
)
b
8
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