The Algebra Teacher\'s Guide to Reteaching Essential Concepts and Skills

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.

1. 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

2. Explain that logarithms may be condensed by applying the properties of logarithms in
   reverse.


3. 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