Apply the product, quotient, and power rules to expand or condense logarithmic expressions.
| Rule | Formula |
|---|---|
| Product | log_b(M·N) = log_b(M) + log_b(N) |
| Quotient | log_b(M/N) = log_b(M) − log_b(N) |
| Power | log_b(Mᵖ) = p · log_b(M) |
Expand means to break a single log into a sum/difference of simpler logs. Condense is the reverse — combine separate log terms into one.
When expanding a combination like log(x²y³/z), apply all three rules: quotient rule first to split numerator and denominator, then product rule on the numerator factors, then power rule to pull exponents out front.
b^m · b^n = b^(m+n) → log_b(b^m · b^n) = m+n
All three log rules come directly from the laws of exponents. The logarithm is the inverse of exponentiation, so exponent arithmetic translates into log arithmetic.
Product rule: Since b^m · b^n = b^(m+n), taking log_b of both sides gives log_b(M·N) = log_b(M) + log_b(N).
Quotient rule: Since b^m / b^n = b^(m−n), we get log_b(M/N) = log_b(M) − log_b(N).
Power rule: Since (b^m)^p = b^(mp), we get log_b(M^p) = p · log_b(M).
One-on-one Algebra 2 tutoring builds real fluency with logarithm properties — we work through your actual homework and tests so the rules become second nature.