Last updated October 5, 2026 · Report an error
Expanding is the reverse operation of condensing: a single logarithm of a product or quotient — like log(2×3) or log(20/4) — can be broken apart into a sum or difference of simpler individual logarithms using the product and quotient rules of logarithms. This skill is a routine part of algebra and pre-calculus coursework on logarithm properties, and it's genuinely useful in practice whenever a formula in a science or engineering context needs to be manipulated algebraically — for instance, isolating a single unknown variable that's buried inside a multi-factor logarithm requires expanding it into separate, individually solvable terms first.
How it works
Enter a base and up to two numerator factors plus one denominator factor, representing an expression like log_b(x × y / z). The calculator applies the product rule, log(xy) = log(x) + log(y), and the quotient rule, log(x/z) = log(x) − log(z), together: it writes out log(factor) for each individual factor, connecting the numerator factors with a plus sign and the denominator factor with a minus sign, producing the fully expanded sum-and-difference form. It then verifies the result numerically by confirming the expanded expression evaluates to the same number as the original combined logarithm.
- Enter base.
- Enter numerator factor 1 (a).
- Enter numerator factor 2 (b), optional.
- Enter denominator factor (c), optional.
- Click Calculate to see your results.
Examples
log(6)
Rewritten as log(2 × 3): expands to log(2) + log(3).
log(20/4)
Using the quotient rule: expands to log(20) − log(4).
log(2 × 3 / 5)
Combining both rules: expands to log(2) + log(3) − log(5).
Who should use it
- Algebra and pre-calculus coursework on logarithm properties.
- Isolating an unknown variable buried inside a multi-factor logarithmic expression.
- Preparing a logarithmic expression for differentiation in calculus.
Industry applications
- Algebra and calculus education
- Engineering and science formulas involving logarithms
Advantages
- Applies both the product and quotient rules together in one consistent expansion.
- Numerically verifies the expanded result matches the original.
Limitations
- Only covers products and quotients — sums and differences inside a logarithm have no general expansion rule.
Common mistakes to avoid
- Splitting log(a+b) into log(a) + log(b) — the product/quotient rules only apply to products and quotients inside the log, never to sums or differences.
- Forgetting to flip the sign from plus to minus when expanding a quotient rather than a product.
- Losing track of which factors belong in the numerator (connected by plus signs) versus the denominator (connected by a minus sign) when reconstructing the expanded form.
Best practices
- Double check whether the original expression is truly a product/quotient (expandable) or a sum/difference (not expandable this way) before applying these rules.
- Verify your expanded result by evaluating both the original and expanded forms numerically — they should match.
- Combine with the power rule, log(xⁿ)=n·log(x), for a fuller expansion when factors include exponents.
Tips
- Need to do the reverse — combine several logs into one? Use the Condense Logarithms Calculator.