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
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.
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.
This is the Product Property of Logarithms .
Solution: Applying the Product Property from Example A, we have:
As you might expect, the
Quotient Property of Logarithms
Intro Problem Revisit
If you rewrite
Simplify the following expressions.
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.
Simplify the following logarithmic expressions.
Expand the following logarithmic functions.
Write an algebraic proof of the Quotient Property. Start with the expression
logax−logayand the equations logax=mand logay=nin your proof. Refer to the proof of the product property in Example A as a guide for your proof.