Mathematics July 22, 2026 · 10 min read

Expanding Logarithms: Breaking Down Complex Expressions

Learn to expand logarithmic expressions into simpler terms. Master product, quotient, and power rules with worked examples.

Try the free calculator

Put these formulas into practice with our instant, step-by-step Expanding Logarithms Calculator.

Open Calculator ›

Expanding logarithms means breaking a single logarithm into multiple terms using the product, quotient, and power rules. This skill simplifies calculus operations and helps solve equations.

The Three Rules

  • Product Rule: log(AB) = log(A) + log(B)
  • Quotient Rule: log(A/B) = log(A) - log(B)
  • Power Rule: log(Aⁿ) = n × log(A)

Worked Example

Expand log₂(8x³/y):

  1. Quotient rule: log₂(8x³) - log₂(y)
  2. Product rule: log₂(8) + log₂(x³) - log₂(y)
  3. Power rule: 3 + 3log₂(x) - log₂(y)

Important Note

There is NO rule for expanding log(A + B). Only log(A × B) and log(A/B) can be expanded.