14.5PureStretch

Laws of Logarithms

The log laws mirror the index laws: multiplication becomes addition, division becomes subtraction, and powers come out as multipliers. They let you combine or split logarithms — essential for solving log equations.

30 min Video by Zeeshan Zamurred Exponentials and Logarithms

Edexcel AS Level Maths: 14.5 Laws of LogarithmsWatch the full walkthrough before the notes below.

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What you'll be able to do

Use the product, quotient and power laws
Combine several logarithms into one
Split a single logarithm into parts
Simplify logarithmic expressions

The three laws

Each log law comes directly from the matching index law (since logs are powers).

lo g_{a} x + lo g_{a} y = lo g_{a} (x y)

Product law: add logs ⟶ multiply.

lo g_{a} x - lo g_{a} y = lo g_{a} (\frac{x}{y})

Quotient law: subtract logs ⟶ divide.

lo g_{a} (x^{k}) = k lo g_{a} x

Power law: the index comes out front.

Combining logs

Use the laws to write several logs as a single one — usually the first step in solving an equation.

1Product and quotient laws:

lo g \frac{6 \times 4}{3}

= lo g 8

Answer

lo g 8

Tip — Use the power law first to clear any coefficients in front of logs, then combine with product/quotient.

Splitting logs

The laws also work in reverse, to break a single log into a sum or difference — useful when differentiating or simplifying.

1Quotient:

lo g x^{2} - lo g y

2Power:

2 lo g x - lo g y

Answer

2 lo g x - lo g y

Formula recap

lo g_{a} x + lo g_{a} y = lo g_{a} (x y)

Product law.

lo g_{a} x - lo g_{a} y = lo g_{a} (x / y)

Quotient law.

lo g_{a} (x^{k}) = k lo g_{a} x

Power law.

Common mistakes to avoid

Writing log(x + y) = log x + log y.

log x + log y = log(xy), not log(x + y).

Bringing a coefficient inside incorrectly.

k·log x = log(xᵏ) — the coefficient becomes a power.

Key takeaways

Product: log x + log y = log(xy).
Quotient: log x − log y = log(x/y).
Power: log(xᵏ) = k log x.
These mirror the index laws and work both ways.

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