The laws of logarithms

Logarithms come with three laws that turn hard operations into easy ones. They're not arbitrary — each one is an exponent rule wearing a disguise.

The three laws

The headline is the product law: logs turn multiplication into addition. That single property is why logarithms were invented — before calculators, they collapsed back-breaking multiplications into simple sums.

Where they come from

The laws are just exponent rules in reverse. Take the product law: write M = bˣ and N = bʸ, so x = logᵦM and y = logᵦN. Then

• M·N = bˣ · bʸ = bˣ⁺ʸ (the exponent rule: multiply → add exponents)
• so logᵦ(M·N) = x + y = logᵦM + logᵦN. ✓

The quotient and power laws follow from the matching exponent rules the same way.

Worked example

Expand log(x³y / z):

1. Quotient law splits the division: log(x³y) − log(z)
2. Product law splits the multiplication: log(x³) + log(y) − log(z)
3. Power law brings the exponent down: 3 log(x) + log(y) − log(z)

Running it backward — condensing — uses the same laws in reverse to fold a sum of logs into a single log.

The change-of-base trick

Your calculator only has log (base 10) and ln (base e). To find log₂(50), rewrite it: logᵦ(x) = ln(x) / ln(b). So log₂(50) = ln(50) / ln(2) ≈ 5.64. Any base, two keystrokes.

Why this matters

Solving exponential equations almost always ends with the power law — it's the step that pulls x down out of the exponent so you can isolate it. In calculus, these laws turn nasty products and quotients into sums you can differentiate term by term.