Teaching Notes 8.21: Using the Properties of Logarithms
The properties of logarithms can be used to express the sum, difference, or product of a number
(or numbers) and a logarithm as a simple logarithm. Many students encounter trouble when they
apply the properties of logarithms, particularly when applying them in reverse.
- Explain the three properties of logarithms toyour students and provide examples. Note that
ais the base of the logarithmic function.a>0,a=1.
First property: logaMN=logaM+logaN
log 216 =log 28 +log 22
Check: 4 = 3 + 1
Second property: logaMN=logaM−logaN
log 2162 =log 216 −log 22
Check: 3 = 4 − 1
Third property: logaMk=klogaM
log 285 =5log 28
log 285 = 5 · 3
Check: 85 = 215
- Explain that logarithms may be condensed by applying the properties of logarithms in
reverse. - Review the information and examples on theworksheet with your students. Note that the
base isa, which can represent any base. Thus, students will never be able to find a specific
number.
EXTRA HELP:
Always apply the third property before applying the first or second property.
ANSWER KEY:
(1)loga 5 (2)loga 36 (3)loga 4 (4)loga 12
------------------------------------------------------------------------------------------
(Challenge)Terri’s method is wrong. She should have simplified
1
2
loga64 as loga 64
1
(^2) or loga 8
------------------------------------------------------------------------------------------first. Then she should have rewritten loga^8 +log^2 2asloga16.
314 THE ALGEBRA TEACHER’S GUIDE