Three rules, and that is all there are. A product becomes a sum, a quotient becomes a difference, and an exponent becomes a coefficient. So log₂(x³y/z²) expands to 3 log₂x + log₂y − 2 log₂z. There is no rule for a sum inside the logarithm.
Why the rules work
The rules are not arbitrary. A logarithm is an exponent, and each rule is an exponent law read backwards.
Multiplying powers of the same base adds their exponents: b^m × b^n = b^(m+n). Take logarithms of both sides and that becomes log(xy) = log x + log y. The product rule is the multiplication law, restated.
Dividing powers subtracts exponents, which gives the quotient rule. Raising a power to a power multiplies exponents, which gives the power rule.
Seeing it that way makes the rules impossible to misremember, and it makes the missing fourth rule obvious: there is no exponent law for b^m + b^n, so there can be no logarithm rule for a sum.
The three rules
Product: log(xy) = log x + log y. Multiplication inside becomes addition outside.
Quotient: log(x/y) = log x − log y. Division becomes subtraction.
Power: log(xⁿ) = n log x. An exponent comes out as a coefficient in front.
All three keep the base unchanged. A change of base is a separate operation with its own formula, and mixing bases inside one expansion is a step that has to be done deliberately.
The power rule covers roots too, since a root is a fractional exponent: log(√x) is ½ log x. That is worth knowing, because it is the version that appears most often in practice.
The rule that does not exist
log(a + b) is NOT log a + log b. This is the single commonest error on the topic, and it is worth stating as bluntly as possible.
Addition inside a logarithm cannot be broken up at all. There is no rule, no identity, and no rearrangement that helps.
A single example settles it. log₁₀(10 + 90) = log₁₀(100) = 2. But log₁₀10 + log₁₀90 = 1 + 1.954 = 2.954. Not close, and not a rounding.
The mistake comes from pattern-matching the product rule, and the fix is to remember why the rules exist: they mirror exponent laws, and there is no exponent law for a sum.
Nor is log(a − b) anything in particular. Only multiplication, division and exponentiation inside a logarithm can be moved outside.
Expanding and condensing
They are the same three rules run in opposite directions, which is why this page shows both forms at once.
Expanding breaks one logarithm into several. It is what you do before differentiating — logarithmic differentiation turns a messy product into a sum, and a sum is far easier to differentiate.
Condensing gathers several into one. It is what you do to solve an equation, because once both sides are a single logarithm with the same base you can drop the logarithms entirely.
One caution on condensing: coefficients must go back inside as exponents first. Writing 2 log x + log y as log(2x · y) is wrong; it is log(x²y).
And a solution obtained by condensing must be checked against the original, because the domain can change — log x + log(x−3) requires x > 3, while log(x²−3x) alone permits negative x as well. Extraneous solutions arise exactly there.
Where it gets used
Solving equations. Condense both sides to a single logarithm and the logarithms cancel, leaving something ordinary.
Calculus. Logarithmic differentiation expands a product into a sum before differentiating, which turns the product rule applied five times into five easy terms.
Statistics. Log-likelihoods turn products of probabilities into sums, which is both numerically safer and analytically easier — multiplying a thousand probabilities underflows to zero, and adding their logarithms does not.
Scales. Decibels, pH and the Richter scale are logarithmic, so combining or comparing values on them uses these rules directly.
Sources and methodology
The logarithm laws are standard; these are the references.
Method. Structured input rather than expression parsing, deliberately: the rules are the subject, and a parser would move the difficulty into the typing. Each output term names the rule that produced it, so the page teaches the derivation rather than only the answer. The base is carried through unchanged, because none of the three rules touches it — mixing bases is a separate operation with its own step. That engine is verified on every change against 99 hand-written assertions, including that a product, quotient and power combined expand to the correct three-term form, and that the page states explicitly that log(a + b) is not log a + log b. The count and the per-case breakdown are published on the formula verification page.
Read the guide
To evaluate a logarithm rather than rearrange one, the Log Calculator handles any base including the change-of-base step.