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Properties of Logarithms – Expanding Logarithms
What are the Properties of Logarithms?
The properties of logarithms are very similar to the properties of exponents because as we have seen before
every exponential equation can be written in logarithmic form and vice versa.
Properties for Expanding Logarithms
There are 5 properties that are frequently used for expanding logarithms. These properties are summarized
in the table below. When applying the properties of logarithms in the examples shown below and in future
examples, the properties will be referred to by number.
To help see where one of the properties comes from let’s look at one of the properties of exponents. If we
start with the zero exponent rule that states a
0 = 1 (or that any number raised to the zero power will equal
one) and we write this property in logarithmic form, we get 0 = log 1a or log 1a = 0.This is property number
1 which says that log of 1 will always equal zero no matter what the base is. If we went through and rewrote
each of the properties of exponents we would get the properties of logarithms shown above.
When we are expanding logarithms there is not a specific order in which these properties must be applied,
but some guidelines are listed below.
Properties for Expanding Logarithms
Property 1: log 1a = 0 – Zero Exponent Rule
Property 2: log aa = 1
Property 3: log (xy)a = log xa + loga y – Product Rule
Property 4: (^) a a a
x log log x log y y
Ê ˆ
Á ˜=^ -
Ë ¯
Property 5:
y log (^) a x = y log (^) ax– Power Rule
Guideline for Expanding Logarithms
∑ Rewrite any radicals using rational exponents (fractions).
∑ Apply Property 3 or 4 to rewrite the logarithm as addition and subtract instead of multiplication and division.
∑ Apply Property 5 to move the exponents out front of the logarithms.
∑ Apply Property 1 or 2 to simplify the logarithms.
and the division as subtraction. Note that 4x means 4 times x
which is why Property 3 has been used to rewrite the
logarithm using addition.
Examples – Now let’s use the properties of logarithms to expand logarithms.
Example 1 : Use the properties of logarithms to expand ( )
2 7 log 3 x y.
2 7 2 7 log 3 x y = log x 3 + log y 3 Use property 3 to rewrite the multiplication as addition.
= 2log x 3 + 7 log y 3 Use Property 5 to move the exponents out front.
Thus, ( )
2 7 log 3 x y = 2log x 3 +7 log y. 3
Example 2 : Use the properties of logarithms to expand (^8 )
x log. y
Ê ˆ
ÁÁ ˜˜
Ë ¯
1 2
(^8 3 )
x x log log y y
Ê ˆ Ê^ ˆ
Á ˜= Á^ ˜
Á ˜ Á ˜
Ë ¯ Ë ¯
Rewrite the radical using rational exponents (fractions).
(^1 ) 2 = log x 8 - log y 8 Use property 4 to rewrite the division as subtraction.
8 8
log x 3log y 2
= - Use Property 5 to move the exponents out front.
Thus, (^8 38 )
x 1 log log 3log y. y 2
Ê ˆ
Á ˜=^ -
Á ˜
Ë ¯
Example 3 : Use the properties of logarithms to expand (^4 )
4x log. y
Ê ˆ
Á ˜
Ë ¯
9 4 9 4 4 4
4x log log 4 log x log y y
Ê ˆ
Á ˜=^ +^ -
Ë ¯
Use properties 3 and 4 to rewrite the multiplication as addition
= log 4 4 + log 4 x - 9log 4 y Use Property 5 to move the exponents out front.
= 1 + log x 4 - 9log 4 y Use Property 2 to simplify the logarithm.
Thus, (^4 4 ) 9
4x log 1 log x 9log y. y
Ê ˆ
Á ˜=^ +^ -
Ë ¯
Addition Examples
If you would like to see more examples of expanding logarithm, just click on the link below.
Additional Examples
Practice Problems
Now it is your turn to try a few practice problems on your own. Work on each of the problems below and
then click on the link at the end to check your answers.
Problem 1 : Use the properties of logarithms to expand
5 3
(^6 )
x log. y
Ê ˆ
Á ˜
Á ˜
Ë ¯
Problem 2 : Use the properties of logarithms to expand (^5)
10 x log. y
Ê ˆ
Á ˜
Á ˜
Ë ¯
Problem 3 : Use the properties of logarithms to expand
14 7
(^8 )
x y log. z
Ê ˆ
Á ˜
Ë ¯
Problem 4 : Use the properties of logarithms to expand ( )
3 8 ln 7x y.
Problem 5 : Use the properties of logarithms to expand (^2 5 )
log. x y
Ê ˆ
Á ˜
Ë ¯
Problem 6 : Use the properties of logarithms to expand
6 5
(^9 3 )
x log. y (z 2)
Ê ˆ
Á ˜
Á - ˜
Ë ¯
Solutions to Practice Problems
