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

Write \(2\log_3 6-\log_3 4\) as a single logarithm and find its value.

\(2\log_3 6=\log_3 36\)
\(\log_3 36-\log_3 4=\log_3\dfrac{36}{4}=\log_3 9\)

Answer: \(\log_3 9=2\)

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
AA SL	In the IB formula booklet
AA HL	In the IB formula booklet
AI HL	In the IB formula booklet
AI SL	Not in the booklet: learn it

From our own IB Maths formula sheets, in our words. The IB gives its booklet to schools, so we checked against our own copy: your teacher has the official one. Official: IB DP Mathematics page.

Practise and revise

Practise AA SL exponentials and logarithms questions
Practise AI SL logarithms questions
Revise Exponents and logs (AA SL)
Revise Logarithms (AA HL)
Tool Graph of y = ln x
Print AA SL 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. IB 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?

AA SL: in the IB formula booklet. AA HL: in the IB formula booklet. AI HL: in the IB formula booklet. AI SL: not in the booklet: learn it. This comes from our own IB 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 IB Math Revision. Not produced or endorsed by the International Baccalaureate Organization.