Properties of Exponents and Logarithms
Exponents
Let
a
and
b
be real numbers and
m
and
n
be integers. Then the following properties of
exponents hold, provided that all of the expressions appearing in a particular equation are
dened.
1.
a
m
a
n
=
a
m
+
n
2. (
a
m
)
n
=
a
mn
3. (
ab
)
m
=
a
m
b
m
4.
a
m
a
n
=
a
m
n
,
a
6
= 0 5.
a
b
m
=
a
m
b
m
,
b
6
= 0 6.
a
m
=1
a
m
,
a
6
= 0
7.
a
1
n
=
n
p
a
8.
a
0
= 1,
a
6
= 0 9.
a
m
n
=
n
p
a
m
=
n
p
a
m
where
m
and
n
are integers in properties 7 and 9.
Logarithms
Denition:
y
= log
a
x
if and only if
x
=
a
y
, where
a >
0.
In other words, logarithms are exponents.
Remarks:
log
x
always refers to log base 10, i.e., log
x
= log
10
x
.
ln
x
is called the natural logarithm and is used to represent log
e
x
, where the irrational
number
e
2
:
71828. Therefore, ln
x
=
y
if and only if
e
y
=
x
.
Most calculators can directly compute logs base 10 and the natural log. For any other
base it is necessary to use the change of base formula: log
b
a
=ln
a
ln
b
or log
10
a
log
10
b
.
Properties of Logarithms
(Recall that logs are only dened for positive values of
x
.)
For the natural logarithm For logarithms base
a
1. ln
xy
= ln
x
+ ln
y
1. log
a
xy
= log
a
x
+ log
a
y
2. ln
x
y
= ln
x
ln
y
2. log
a
x
y
= log
a
x
log
a
y
3. ln
x
y
=
y
ln
x
3. log
a
x
y
=
y
log
a
x
4. ln
e
x
=
x
4. log
a
a
x
=
x
5.
e
ln
x
=
x
5.
a
log
a
x
=
x
Useful Identities for Logarithms
For the natural logarithm For logarithms base
a
1. ln
e
= 1 1. log
a
a
= 1, for all
a >
0
2. ln 1 = 0 2. log
a
1 = 0, for all
a >
0
1
