Laws of logarithms: log xy = log x + log y

The laws of logarithms are log xy = log x + log y; log(x/y) = log x − log y; log xᵐ = m log x.

\(\displaystyle \log_a xy=\log_a x+\log_a y\) \(\displaystyle \log_a\frac{x}{y}=\log_a x-\log_a y\) \(\displaystyle \log_a x^m=m\log_a x\)

What each letter means

\(a\) the base (positive, not 1)
\(x,\ y\) positive numbers
\(m\) any power

When to use it

To combine several logs into one, to split one log into simpler parts, or to bring a power down so you can solve an equation such as 3ˣ = 20.

Worked example

Solve \(\log_2(x+3)+\log_2(x-3)=4\).

\(\log_2\big((x+3)(x-3)\big)=4\), so \(x^2-9=2^4=16\)
\(x^2=25\), so \(x=\pm5\); reject \(x=-5\) because \(\log_2(x-3)\) needs \(x>3\)

Answer: \(x=5\)

Common mistake

Writing log(x + y) = log x + log y. The product law is about log(xy); there is no law for log(x + y).

On your course

Course	In the exam
Edexcel A Level (9MA0)	Not in the Edexcel booklet: learn it
Edexcel International A Level	Not in the booklet: learn it
AQA or OCR A Level	Check your board’s formula booklet: AQA formulae booklet (PDF) · OCR A Level Maths assessment page

From our own A Level Maths formula sheets, in our words. Official: Pearson's formulae booklet (PDF) · Pearson's IAL formulae booklet (PDF).

Practise and revise

Practise A Level Maths practice
Revise Exponentials and logarithms
Revise IAL P2 logs and exponentials
Print Edexcel A Level (9MA0) one-page formula sheet

Laws of logarithms formula card: log xy = log x + log y; log(x/y) = log x − log y; log xᵐ = m log x. A Level Math Revision — Laws of logarithms formula card. Download the card (PNG) to save or print it.

Questions

What is the laws of logarithms formula?

The laws of logarithms are log xy = log x + log y; log(x/y) = log x − log y; log xᵐ = m log x. a: the base (positive, not 1); x, y: positive numbers; m: any power.

Is the laws of logarithms given in the exam?

Edexcel A Level (9MA0): not in the Edexcel booklet: learn it. Edexcel International A Level: not in the booklet: learn it. AQA or OCR A Level: check your board’s formula booklet. This comes from our own A Level Maths formula sheets; your teacher has the official booklet.

Is log(x + y) equal to log x + log y?

No. log x + log y = log(xy). A sum inside a log cannot be split.

Our own wording, examples and card, checked by A Level Math Revision. Not produced or endorsed by Pearson Edexcel, AQA or OCR.