Mathematics Assistance Centre
A unit within the Centre for Student Success
www.wlu.ca/mac
Reference Sheet:
Example: Express as a single logarithm.
Laws of Logarithms
2 log3 7
log3 5
= log3 72
The general form of a logarithmic expression is
loga m, where a is the base of the logarithm and
m is its argument. Logarithms can be expressed
in exponential form, where m = ay is equivalent
to y = loga m, with m > 0, a > 0 and a 6= 1.
A logarithm of base 10 is referred to as the "common
logarithm", and is often denoted as log m.
A logarithm of base e is known as the "natural
logarithm" and is denoted as ln m.
Examples:
log2 8 = 3 because we know 23 = 8
log9 3 =
log
1
1000
1
2
since 91=2 = 3
=
3 since 10
3
=
1
1000
log3 5
49
5
= log3
Example: Expand the expression.
log6
3
p
4 5
p
log6 4 5
= log6 3
log6 4 + log6 51=2
= log6 3
= log6 3
log6 4
1
2
log6 5
Example: Evaluate e3 ln 4
e3 ln 4 = eln 4
3
= eln 64
For appropriate values for the base and argument,
we have the following properties.
Basic Properties:
loga 1 = 0
loga a = a
= 64
Example: Simplify log16 32
Recognizing that both 16 and 32 can be
expressed as an exponential with base 2, we
use the change of base formula for logarithms:
log16 32 =
log2 32
5
=
log2 16
4
Laws of Logarithms:
loga (mn) = loga m + loga n
m
loga
= loga m loga n
n
loga (mp ) = p loga m
loga (ap ) = p
aloga m = m
Change of Base Formula:
logb m
ln m
loga m =
=
logb a
ln a
Example:
Evaluate log5 20, accurate to 3 decimals.
Many calculators can only evaluate logarithms
of base 10 or base e, so the change of base
formula is helpful:
log5 20 =
ln 20
=
ln 5
2:995732:::
1:609437:::
:
= 1:861
Note: The logarithmic operation is not linear;
that is,
loga (m + n) 6= loga m + loga n
loga (m
n) 6= loga m
loga n