Lesson 5.11
Product Property of Logarithms
The log of a product equals the sum of the logs. This property turns multiplication inside a log into addition outside — the fundamental trick that makes logarithms powerful.
Introduction
Before calculators, logarithms were invented precisely because they turn multiplication into addition. This property comes directly from the exponent rule .
Past Knowledge
Log definition (5.6), exponent product rule ().
Today's Goal
Apply the product property to expand and combine logs.
Future Success
Combined with quotient (5.12) and power (5.13), this forms the toolkit for 5.14 and solving equations.
Key Concepts
Product Property
The log of a product = sum of the logs (same base)
Why It Works
Let and . Then , so .
⚠️ Only for Products INSIDE
There is NO property for the log of a sum.
Worked Examples
Example 1: Expanding
BasicExpand .
Verify: ✓
Example 2: Combining
IntermediateWrite as a single log: .
Example 3: With Variables
AlgebraicExpand .
Common Pitfalls
Log of a Sum ≠ Sum of Logs
. The left is ; the right is .
Different Bases
The property only works when all logs have the same base. cannot be combined.
Real-Life Applications
This is how slide rules worked for centuries — engineers added logarithmic scales to multiply numbers. It's also why decibels add up: combining two 50 dB sounds doesn't give 100 dB; it gives about 53 dB (log of the sum of intensities).
Practice Quiz
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