Properties of Exponents and Logarithms
[Pages:1]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.
mn
aa
=
m+n
a
2.
(am)n
=
mn
a
3.
(ab)m
=
mm
ab
m
T 4.
a
an
=
m
a
n, a = 0
p 7.
1 an
=
n
a
T 5.
a m
b
=
m
a
bm
,
b
=
0
T 8.
0
a
=
1,
a
=
0
T 6.
a
m=
1
am
,
a
=
0
p p 9.
m an
=
n am =
n ?m a
where m and n are integers in properties 7 and 9.
Logarithms
Denition:
y
=
loga x
if
and
only
if
x
=
y
a
,
where
a
>
0.
In other words, logarithms are exponents.
Remarks:
log x always refers to log base 10, i.e., log x = log10 x.
ln x is called
number e %
the natural logarithm 2:71828. Therefore, ln
and x=
is
y
used to represent loge
if
and
only
if
y
e
=
x.
x,
where
the
irrational
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:
logb a
=
ln a ln b
or
log10 log10
a b
.
Properties of Logarithms (Recall that logs are only dened for positive values of x.)
For the natural logarithm
1. ln xy = ln x + ln y
2. ln x = ln x ln y
? y
3.
ln
y
x
=
y
ln x
4.
ln
x
e
=
x
5.
ln x
e
=
x
For logarithms base a
1. loga xy = loga x + loga y
2. loga x = loga x loga y
? y
3.
loga
y
x
=
y
loga x
4. 5.
laologgaa
x
a
=
x
x=x
Useful Identities for Logarithms
For the natural logarithm For logarithms base a
1. ln e = 1 2. ln 1 = 0
1. loga a = 1, for all a > 0 2. loga 1 = 0, for all a > 0
1
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