Lesson 5.14
Expanding & Condensing Expressions
Now we combine all three properties — product, quotient, and power — to fully expand a complex log into simple terms or to condense multiple logs into one.
Introduction
Expanding means breaking one log into several. Condensing means combining several into one. Both directions use the same three properties — just in reverse order.
Past Knowledge
Product (5.11), quotient (5.12), power (5.13) properties.
Today's Goal
Fully expand and condense multi-step log expressions.
Future Success
Condensing logs into one is needed to solve log equations in 5.17.
Key Concepts
Expanding (one → many)
1. Quotient → subtraction
2. Product → addition
3. Power → coefficient
Condensing (many → one)
1. Coefficient → power (move up)
2. Addition → product (combine)
3. Subtraction → quotient (combine)
Worked Examples
Example 1: Full Expansion
ExpandingExpand .
Quotient
Product
Power
Example 2: Full Condensing
CondensingCondense .
Coefficients → powers
Addition → product
Subtraction → quotient
Example 3: Expanding with Roots
AdvancedExpand .
Quotient
Product
Power (rewrite )
Common Pitfalls
Order Matters When Condensing
Always move coefficients up as powers first, then combine with product/quotient. Combining first can give wrong groupings.
Distributing a Coefficient Incorrectly
, not . Apply the coefficient to each term if distributing.
Real-Life Applications
Many scientific formulas involve products, quotients, and powers inside logs. Expanding lets you isolate the variable you care about; condensing is required before solving log equations.
Practice Quiz
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