Lesson 5.12
Quotient Property of Logarithms
The log of a quotient equals the difference of the logs. Just as the product property mirrors , this mirrors .
Introduction
The quotient property is the "subtraction version" of the product property. Together, they let you break apart complex log expressions into simpler pieces — or combine simple pieces into one log.
Past Knowledge
Product property (5.11), exponent quotient rule.
Today's Goal
Apply the quotient property to expand and combine logs.
Future Success
5.14 uses product + quotient + power together to fully expand/condense expressions.
Key Concepts
Quotient Property
The log of a quotient = difference of the logs (same base)
Complete Property Set So Far
Product:
Quotient:
⚠️ Direction Matters
, not
The first term goes in the numerator, the subtracted term in the denominator.
Worked Examples
Example 1: Expanding
BasicExpand .
Verify: ✓
Example 2: Combining
IntermediateWrite as a single log: .
Example 3: With Variables
AlgebraicExpand using product and quotient properties.
(The will simplify further with the Power Property in 5.13)
Common Pitfalls
"Cancelling" the Logs
. The quotient property applies when dividing inside one log, not when dividing two separate logs.
Subtraction Order
. If you reverse it, — a completely different result!
Real-Life Applications
The quotient property explains why logarithmic scales measure ratios. The difference in decibels between two sounds equals — a direct application of the quotient property.
Practice Quiz
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