Expanding Logarithms Calculator
Expand a single logarithm into multiple logarithmic terms
Enter expressions like: log(a×b), log(a÷b), log(a^n), or log(a×b÷c)
How to Use This Calculator
Enter Logarithmic Expression
Type the logarithmic expression you want to expand. Examples: log(2×3), log(8÷2), log(4^3), or log(2×3÷6).
Specify the Base
Enter the base of the logarithm (default is 10 for common logarithm). Use e ≈ 2.71828 for natural logarithms.
Expand
Click "Expand Logarithms" to break down the single logarithm into multiple logarithmic terms using logarithm properties.
Formulas
log(a × b) = log(a) + log(b)
Example: log(2 × 3) = log(2) + log(3)
Product rule: multiplication becomes addition
log(a ÷ b) = log(a) - log(b)
Example: log(8 ÷ 2) = log(8) - log(2)
Quotient rule: division becomes subtraction
log(an) = n × log(a)
Example: log(2³) = 3 × log(2)
Power rule: exponent becomes coefficient
log(a × b ÷ c) = log(a) + log(b) - log(c)
Example: log(2 × 3 ÷ 6) = log(2) + log(3) - log(6)
Combining multiple rules
About Expanding Logarithms Calculator
The Expanding Logarithms Calculator helps you break down a single logarithmic expression into multiple logarithmic terms. This process uses the fundamental properties of logarithms: the product rule, quotient rule, and power rule. Expanding logarithms is the inverse operation of condensing logarithms and is essential for solving logarithmic equations and simplifying calculations.
When to Use This Calculator
- Solving Logarithmic Equations: Expand logarithms to isolate variables
- Simplifying Calculations: Break down complex logarithms for easier computation
- Differentiation: Expand logarithms before taking derivatives
- Pre-Calculus Problems: Work with logarithmic identities and properties
- Mathematical Analysis: Prepare expressions for further mathematical operations
Why Use Our Calculator?
- ✅ Instant Expansion: Expand logarithmic expressions instantly
- ✅ Multiple Rules: Handles product, quotient, and power rules automatically
- ✅ Step-by-Step: See which logarithm properties are being applied
- ✅ Educational: Learn how to expand logarithms through examples
- ✅ 100% Free: No registration or payment required
- ✅ Accurate: Uses correct logarithm properties for reliable results
Common Applications
Calculus: When differentiating log(x²), you can expand first: log(x²) = 2log(x), then the derivative is easier: d/dx[2log(x)] = 2/x.
Algebra: When solving log(2x) = 3, you can expand: log(2) + log(x) = 3, so log(x) = 3 - log(2), then x = 10^(3 - log(2)).
Pre-Calculus: Simplify complex logarithmic expressions to prepare for graphing or finding key features of logarithmic functions.
Tips for Best Results
- Remember: log(a × b) = log(a) + log(b) - product becomes sum
- Remember: log(a ÷ b) = log(a) - log(b) - quotient becomes difference
- Remember: log(a^n) = n × log(a) - exponent becomes coefficient
- You can combine multiple rules: log(2 × 3² ÷ 6) = log(2) + 2log(3) - log(6)
- Expanding is the opposite of condensing - they reverse each other
Frequently Asked Questions
What does "expanding logarithms" mean?
Expanding logarithms means breaking down a single logarithmic expression into multiple logarithmic terms using the properties of logarithms (product rule, quotient rule, and power rule).
How do I expand log(2×3)?
Using the product rule: log(2 × 3) = log(2) + log(3). Multiplication inside the logarithm becomes addition outside.
How do I expand log(8÷2)?
Using the quotient rule: log(8 ÷ 2) = log(8) - log(2). Division inside the logarithm becomes subtraction outside.
How do I expand log(4³)?
Using the power rule: log(4³) = 3 × log(4). The exponent inside the logarithm becomes a coefficient outside.
What's the difference between expanding and condensing logarithms?
Expanding breaks one logarithm into multiple (e.g., log(6) → log(2) + log(3)). Condensing combines multiple logarithms into one (e.g., log(2) + log(3) → log(6)). They are inverse operations.