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Product and Quotient Properties of Logarithms

Combine and expand logarithmic expressions

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Practice Product and Quotient Properties of Logarithms
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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 log3162log32 . "This is how much each of you owes," he says as he drops the bill on the table. How much do each of you owe?


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 logbx+logby .

Solution: First, notice that these logs have the same base. If they do not, then the properties do not apply.

logbx=m and logby=n , then bm=x and bn=y .

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 m=logbx and n=logby , therefore logbxy=logbx+logby .

This is the Product Property of Logarithms .

Example B

Expand log124y .

Solution: Applying the Product Property from Example A, we have:


Example C

Simplify log315log35 .

Solution: As you might expect, the Quotient Property of Logarithms is logbxy=logbxlogby (proof in the Problem Set). Therefore, the answer is:


Intro Problem Revisit

If you rewrite log3162log32 as log31622 , you get log381 .

34=81 so you each owe $4.

Guided Practice

Simplify the following expressions.

1. log78+log7x2+log73y

2. logylog20+log8x

3. log232log2z

4. log816xy2


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 log816 , use the definition and powers of 2: 8n=1623n=243n=4n=43 .

Explore More

Simplify the following logarithmic expressions.

  1. log36+log3ylog34
  2. log12logx+logy2
  3. log6x2log6xlog6y
  4. ln8+ln6ln12
  5. ln7ln14+ln10
  6. log1122+log115log1155

Expand the following logarithmic functions.

  1. log6(5x)
  2. log3(abc)
  3. log(a2b)
  4. log9(xy5)
  5. log(2xy)
  6. log(8x215)
  7. log4(59y)
  8. Write an algebraic proof of the Quotient Property. Start with the expression logaxlogay and the equations logax=m and logay=n in your proof. Refer to the proof of the product property in Example A as a guide for your proof.


Product Property of Logarithms

Product Property of Logarithms

The product property of logarithms states that as long as b \ne 1, then \log_b xy=\log_b x + \log_b y
Quotient Property of Logarithms

Quotient Property of Logarithms

The quotient property of logarithms states that as long as b \ne 1, then \log_b \frac{x}{y}=\log_b x - \log_b y.

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