8.8: Product and Quotient Properties of Logarithms
Your friend Robbie works as a server at a pizza parlor. You and two of your friends go to the restaurant and order a pizza. You ask Robbie to bring you separate checks so you can split the cost of the pizza. Instead of bringing you three checks, Robbie brings you one with the total
Guidance
Just like exponents, logarithms have special properties, or shortcuts, that can be applied when simplifying expressions. In this lesson, we will address two of these properties.
Example A
Simplify
Solution: First, notice that these logs have the same base. If they do not, then the properties do not apply.
Now, multiply the latter two equations together.
Recall, that when two exponents with the same base are multiplied, we can add the exponents. Now, reapply the logarithm to this equation.
Recall that
This is the Product Property of Logarithms.
Example B
Expand
Solution: Applying the Product Property from Example A, we have:
Example C
Simplify
Solution: As you might expect, the Quotient Property of Logarithms is
Intro Problem Revisit
If you rewrite
Guided Practice
Simplify the following expressions.
1.
2.
3.
4.
Answers
1. Combine all the logs together using the Product Property.
2. Use both the Product and Quotient Property to condense.
3. Be careful; you do not have to use either rule here, just the definition of a logarithm.
4. When expanding a log, do the division first and then break the numerator apart further.
To determine
Vocabulary
 Product Property of Logarithms

As long as
b≠1 , thenlogbxy=logbx+logby
 Quotient Property of Logarithms

As long as
b≠1 , thenlogbxy=logbx−logby
Practice
Simplify the following logarithmic expressions.

log36+log3y−log34 
log12−logx+logy2 
log6x2−log6x−log6y 
ln8+ln6−ln12 
ln7−ln14+ln10 
log1122+log115−log1155
Expand the following logarithmic functions.

log6(5x) 
log3(abc) 
log(a2b) 
log9(xy5) 
log(2xy) 
log(8x215) 
log4(59y)  Write an algebraic proof of the Quotient Property. Start with the expression
logax−logay and the equationslogax=m andlogay=n in your proof. Refer to the proof of the product property in Example A as a guide for your proof.
Image Attributions
Here you'll use and apply the product and quotient properties of logarithms.