Who this is for: Cambridge IGCSE Mathematics (0580/0607) Extended students who want logarithms — laws of logs, solving equations and links to indices — to become a reliable source of marks instead of a topic they only half-remember.
What query it owns: how to understand and revise logarithms in Cambridge IGCSE Mathematics.
Why this is safe: this page owns the logarithms revision-guide angle, while Tutopiya’s Logarithms subtopic page owns the learning resource and the free logarithms quiz owns the practice.

Logarithms are the inverse of indices. In Cambridge IGCSE Mathematics (0580/0607) Extended you use them to solve equations like 2ˣ = 10, simplify expressions using log laws, and interpret answers in context. The notation feels unfamiliar at first, but the mark schemes follow predictable patterns. This guide explains the subtopic, the command words on papers, and where to practise.

Key takeaways

• logₐ b = c means aᶜ = b — the logarithm is the power you raise a to get b.
• The three key laws: log(xy) = log x + log y; log(x/y) = log x − log y; log(xⁿ) = n log x.
• To solve aˣ = k, take logs of both sides or rewrite using the same base.
• ln is log base e; calculators provide log and ln buttons — know which the question expects.

What are logarithms in Cambridge IGCSE Maths?

A logarithm answers the question: to what power must the base be raised to obtain a given number? If aˣ = b, then x = logₐ b. In Cambridge IGCSE Mathematics (Extended) you evaluate simple logarithms, apply the laws of logarithms to simplify expressions, and solve exponential equations using logs.

Read the worked examples on Tutopiya’s Logarithms subtopic page before attempting questions.

The core ideas you must master

Idea	What it means	How the exam uses it
Definition	logₐ b = c ↔ aᶜ = b	”Write log₂ 8 = 3 as a power of 2”
Product law	log(xy) = log x + log y	”Simplify log 4 + log 5”
Quotient law	log(x/y) = log x − log y	”Simplify log 20 − log 4”
Power law	log(xⁿ) = n log x	”Write 3 log 2 as a single log”
Solving aˣ = k	Take logs of both sides	”Solve 3ˣ = 7, giving x correct to 3 d.p.”

How to solve an equation using logarithms — step by step

1. Rewrite the equation so the unknown is in the index: e.g. 2ˣ = 15.
2. Take logs of both sides (log or ln — be consistent).
3. Apply the power law to bring the index down: x log 2 = log 15.
4. Rearrange to make the unknown the subject: x = log 15 ÷ log 2.
5. Calculate with a calculator; round only at the end if decimal places are requested.

Confirm the method with the free Logarithms quiz.

Log laws at a glance

Law	Formula	Exam use
Product	log(xy) = log x + log y	Combine sums of logs
Quotient	log(x/y) = log x − log y	Split or combine differences
Power	log(xⁿ) = n log x	Bring indices in front
Special values	logₐ 1 = 0; logₐ a = 1	Simplify quickly

Logarithms in past-paper wording: command words that matter

Command word / phrase	What the question wants	Typical stem
Write … as a single logarithm	Apply product/quotient/power laws	”Write log 6 + log 5 as a single logarithm.”
Solve the equation	Use logs to find the unknown in the index	”Solve 5ˣ = 12, giving x correct to 2 decimal places.”
Without using a calculator	Use log laws and known values	”Find log₂ 16 without using a calculator.”
Show that	Prove a log identity	”Show that log 20 − log 4 = log 5.”
Given that log 2 = 0.301…	Use supplied values	”Find log 8, giving your answer to 3 decimal places.”

Worked exam-style stems (how to answer the wording)

1. “Solve 2ˣ = 50, giving your answer correct to 2 decimal places.” x log 2 = log 50 → x = log 50 ÷ log 2 ≈ 5.64 (2 d.p.). Reward: taking logs, rearranging, correct rounding.
2. “Write 2 log 3 + log 4 as a single logarithm.” log 3² + log 4 = log 9 + log 4 = log 36. Reward: power law then product law.
3. “Show that log₁₀ 100 = 2.” 10² = 100, so log₁₀ 100 = 2. Reward: linking definition to powers.

Work the full set on the Functions topical past papers.

How logarithms connect to the wider syllabus

Logarithms are the inverse of Exponents and Surds in Number and support Exponential Growth and Decay. The Cambridge IGCSE Maths resource hub links Functions and Number revision.

Common mistakes students make

• Confusing log(xy) with log x × log y — it is log x + log y.
• Forgetting to take log of the entire side, not just the term with the unknown.
• Using log(a + b) ≠ log a + log b — the product law does not apply to sums inside the log.
• Rounding too early in multi-step calculator work.
• Mixing log (base 10) and ln (base e) in the same equation inconsistently.

When you need more support

If logarithm questions keep costing marks, revisit Exponents and Surds, retake the Logarithms quiz, and ask a Cambridge IGCSE Maths tutor to stabilise the laws.

Frequently asked questions

What is a logarithm in simple terms? The power you raise the base to in order to get a given number. If 2³ = 8, then log₂ 8 = 3.

Is logarithms Core or Extended? Logarithms are examined on the Extended syllabus for Cambridge IGCSE Mathematics (0580/0607).

What is the difference between log and ln? log usually means base 10; ln means base e (≈ 2.718). Your calculator has separate buttons for each.

How do I revise logarithms effectively? Learn the three laws, practise rewriting sums as single logs, then solve exponential equations. Use the logarithms quiz after each stage.

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