Logarithm Properties

Core Idea

Logarithm properties mirror exponent rules: Product Rule: log_b(MN) = log_b(M) + log_b(N). Quotient Rule: log_b(M/N) = log_b(M) - log_b(N). Power Rule: log_b(M^p) = p*log_b(M). Change of Base: log_b(x) = log_a(x)/log_a(b). These properties allow expansion, condensation, and evaluation of logarithmic expressions and are essential for solving exponential and logarithmic equations.

How It's Best Learned

Derive each property from the corresponding exponent rule. Practice expanding single logarithms into sums/differences and condensing sums/differences into single logarithms. Use the change of base formula to evaluate logarithms with unusual bases on a calculator. Give problems that require multiple properties in combination.

Common Misconceptions

• Applying the product rule to log(M + N) (there is no simplification for log of a sum).
• Thinking log(M/N) = log(M)/log(N) (the quotient rule gives subtraction, not division of logs).
• Confusing log(M^p) = p*log(M) with (log M)^p.
• Forgetting that these properties require M, N > 0.

Explainer

Logarithm properties are not arbitrary rules to memorize — they are the direct mirror images of exponent rules you already know. If you understand why exponent rules work, logarithm properties follow naturally.

Recall that exponents have a product rule: b^m · bⁿ = b^m+n. Logarithms are exponents (log_b(x) asks "what exponent gives x?"), so taking the log of a product M · N means adding the exponents that produce M and N: log_b(MN) = log_b(M) + log_b(N). The quotient rule follows the same logic in reverse: b^m / bⁿ = b^m-n, so log_b(M/N) = log_b(M) − log_b(N). And the power rule (b^m)^p = b^mp translates to log_b(M^p) = p · log_b(M) — the exponent moves in front as a multiplier.

The most important thing to notice is what these rules do NOT say. The product rule applies to log(M · N), not to log(M + N). There is no simplification for the logarithm of a sum. This is the most common error in logarithm work — writing log(x + 3) = log(x) + log(3) when that transformation is simply invalid. Whenever you see a sum or difference inside a logarithm, stop: no rule applies.

These properties let you do two powerful things: expand and condense. Expanding means rewriting a single logarithm as a sum or difference (useful for solving equations): log(x²y/z) = 2log(x) + log(y) − log(z). Condensing means combining a sum or difference into a single logarithm (also useful for solving): 2log(x) + log(y) − log(z) = log(x²y/z). Problems may ask for either direction, so practice recognizing which form you're starting with and which you're trying to reach.

The change-of-base formula, log_b(x) = log(x)/log(b), is a practical tool rather than a structural property. It lets you evaluate any logarithm on a calculator that only has log base 10 or ln (natural log). For example, log₅(125) = log(125)/log(5) = 2.097/0.699 = 3. (You can verify: 5³ = 125.) When solving exponential equations with unusual bases — like 3^x = 50 — the change-of-base formula is often the final step that produces a numerical answer.