Log rules (the laws of logarithms)

The log rules, also called the laws of logarithms, turn multiplication into addition and powers into multiplication. The three main ones are: log(xy) = log x + log y, log(x/y) = log x − log y and log(xᵏ) = k log x. They work for logs in any base, including natural logs, ln, so the same rules let you simplify log expressions and solve equations where the unknown is in a power.

What is a logarithm?

A logarithm answers the question: what power do I raise the base to, to get this number? Since 2⁵ = 32, the log of 32 in base 2 is 5.

On a calculator, log usually means base 10 and ln means base e, where e ≈ 2.718. You can only take the log of a positive number.

All the log rules in one table

In the first three rules, every log must have the same base. All three come straight from the exponent rules, because a logarithm is an exponent: multiplying powers means adding exponents, so multiplying numbers means adding their logs. The exponent rules guide shows the matching rule for each one.

Why the product law works

Let logₐ x = m and logₐ y = n, so x = aᵐ and y = aⁿ. Then:

Taking logs base a of both sides gives logₐ(xy) = m + n = logₐ x + logₐ y. The quotient law follows in the same way from aᵐ ÷ aⁿ = aᵐ⁻ⁿ, and the power law from (aᵐ)ᵏ = aᵐᵏ.

Ln rules (natural log rules)

ln x is the logarithm in base e. It follows exactly the same rules, plus two you use all the time:

The graph of y = ln x shows those special values. Here x is the input and y is ln x.

The curve crosses the x-axis at x = 1, because ln 1 = 0, and reaches 1 at x = e. It only exists for x > 0, which is why you cannot take the log of zero or a negative number. Each time x is multiplied by e, y goes up by 1: that is the product law in picture form.

Log rules examples

Example 1: write as a single logarithm

Write 2 log 3 + log 4 − log 6 as a single logarithm.

Use the power law first, then the product and quotient laws:

Example 2: expand a logarithm

Write log(x³y ÷ z²) in terms of log x, log y and log z.

Example 3: evaluate without a calculator

Find log₂ 32, log₃(1/9) and log₈ 4.

• log₂ 32 = 5, because 2⁵ = 32.
• log₃(1/9) = −2, because 3⁻² = 1/9.
• log₈ 4: write both as powers of 2. 8 = 2³ and 4 = 2², so 8^(2/3) = 4 and log₈ 4 = 2/3.

Example 4: solve an equation with the unknown in the power

Solve 5ˣ = 40, giving x to 3 significant figures.

Take logs of both sides, then use the power law to bring x down:

Check: 5² = 25 and 5³ = 125, so an answer between 2 and 3 makes sense.

Example 5: solve an equation with e

Solve e²ˣ = 7.

Take ln of both sides: 2x = ln 7, so x = ½ ln 7 = 0.973 to 3 significant figures.

Example 6: combine logs, then solve

Solve log₂(x + 3) + log₂(x − 3) = 4.

Combine with the product law, then undo the log:

So x² = 25 and x = ±5. But log₂(x − 3) needs x − 3 > 0, so x = −5 is not allowed. The only solution is x = 5.

Example 7: ln on both sides

Solve ln(x + 1) + ln(x − 1) = ln 8.

The left side is ln[(x + 1)(x − 1)] = ln(x² − 1). If ln(x² − 1) = ln 8, then x² − 1 = 8, so x = ±3. Only x = 3 keeps both logs defined, so x = 3.

Here is a quick check before the mistakes.

Common mistakes

• Splitting a sum. log(x + y) is not log x + log y. The product law is about log(xy).
• Multiplying logs. log x × log y is not log(xy). There is no rule for multiplying two logs.
• Dividing logs. log x ÷ log y is not log(x/y). log x ÷ log y is a change of base: it equals log_y x.
• Squaring the log instead of the number. (log x)² is not 2 log x. The power law needs the power inside: log(x²) = 2 log x.
• Keeping invalid solutions. Check every answer in the original equation. The log of zero or a negative number is undefined.
• Mixing bases. The product, quotient and power laws only combine logs with the same base.

Logs in science

Logarithmic scales compress huge ranges of numbers. In chemistry, pH = −log₁₀[H⁺], so each step of 1 on the pH scale is a factor of 10 in hydrogen ion concentration. The decibel scale for sound and the Richter scale for earthquakes work the same way. In A-level Maths and Physics, plotting log y against log x, or ln y against x, turns a power or exponential relationship into a straight line, which lets you find the constants from the gradient and intercept.

Using log rules to get a straight line (A-level)

If two quantities are linked by y = axⁿ, taking logs of both sides and using the product and power laws gives:

That has the form Y = mX + c, with Y = log y and X = log x. So a graph of log y against log x is a straight line with gradient n and y-intercept log a. For example, if the line has gradient 1.5 and crosses the vertical axis at 0.3, then n = 1.5 and a = 10^0.3 = 2.0, so y = 2.0x^1.5.

For an exponential model, y = abˣ, the same steps give log y = x log b + log a. Plot log y against x itself: the gradient is log b and the intercept is log a.

Are the log rules in the formula booklet?

Not the main laws. This was checked against the A-level Maths booklets in November 2026. Edexcel's booklet, under logarithms and exponentials, gives the change of base formula, logₐ x = log_b x ÷ log_b a, and e^(x ln a) = aˣ. The product, quotient and power laws are not printed. AQA's booklet does not include a logarithms section at all. So learn the laws of logs by heart for A-level Maths. The Logarithm Calculator evaluates logs in any base and shows the rules at work, so you can check your answers.

Practise logs in a short timed paper

A-level Maths: Algebra and Functions gives you a short timed paper every day, with 8 questions worth 25 to 35 marks, and exponentials and logarithms come up regularly. In the US, the same practice opens as AP Precalculus. The Graphing Functions course covers exponential growth and decay curves, the graphs that logarithms undo.

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Maths · AlgebraWhat is a logarithm? Logs explained with examplesWhat a logarithm is, how to read and work out logs, log base 10 and natural logs, the log rules and why they work, solving equations and real uses.