Back to course
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 Exponentials and Logarithms
Edexcel AS Level Maths: 14.5 Laws of LogarithmsCurated by Kepler, not produced by us. Watch it before the notes below.
Open on YouTube

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
1

The three laws

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

log⁡ax+log⁡ay=log⁡a(xy)\log_a x + \log_a y = \log_a (xy)
Product law: add logs ⟶ multiply.
log⁡ax−log⁡ay=log⁡a ⁣(xy)\log_a x - \log_a y = \log_a \!\left(\tfrac{x}{y}\right)
Quotient law: subtract logs ⟶ divide.
log⁡a(xk)=klog⁡ax\log_a (x^{k}) = k\log_a x
Power law: the index comes out front.
2

Combining logs

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

Example 1Write log⁡6+log⁡4−log⁡3\log 6 + \log 4 - \log 3 as a single logarithm.
1Product and quotient laws: log⁡6×43\log \dfrac{6 \times 4}{3}.
2=log⁡8= \log 8.
Answerlog⁡8\log 8

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

3

Splitting logs

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

Example 1Expand log⁡ ⁣(x2y)\log\!\left(\dfrac{x^2}{y}\right).
1Quotient: log⁡x2−log⁡y\log x^2 - \log y.
2Power: 2log⁡x−log⁡y2\log x - \log y.
Answer2log⁡x−log⁡y2\log x - \log y

Formula recap

log⁡ax+log⁡ay=log⁡a(xy)\log_a x + \log_a y = \log_a(xy)
Product law.
log⁡ax−log⁡ay=log⁡a(x/y)\log_a x - \log_a y = \log_a(x/y)
Quotient law.
log⁡a(xk)=klog⁡ax\log_a(x^{k}) = k\log_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.

Test yourself

Answer a few questions, then review anything you missed.

Next step

Check your understanding

Work through questions on Laws of Logarithms. Use the feedback to decide what needs another look.

6 past-paper questions · 11 marks

Practise this topic

Related lessons

All 8 lessons

Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.