AQA A-Level Mathematics Paper 1, June 2025: Question 3

1 mark · Easy difficulty · Short Answer

Express the difference of logarithms log base 3 of 2x minus log base 3 of x as a single logarithm.

Practise this question

Question

Question 3 asks to express log base 3 of 2x minus log base 3 of x as a single logarithm and circle the correct answer out of four options: log base 3 of 1/2, log base 3 of 2, log base 3 of x, and log base 3 of 2x squared. The question is worth 1 mark.
Question text

3 Express

log3 2x – log3 x

as a single logarithm.

Circle your answer.

[1 mark]

log3 log3 2 log3 x log3 2x

Mark scheme

Show the mark scheme Mark scheme table for Question 3 shows that 1 mark (B1, AO 1.1b) is awarded for circling the second option, with typical solution log base 3 of 2.

Q Marking instructions AO Marks Typical solution

3 Circles second option 1.1b B1 log 23

Question 3 Total 1

How to answer it

Laws of Logarithms: Simplifying Subtraction

📌 What This Question Tests

This question assesses your foundational algebraic fluency with the laws of logarithms, specifically:

  • Recognising and applying the quotient rule for logarithms: logₐ A − logₐ B = logₐ (A / B) .
  • Simplifying an algebraic fraction inside a logarithm ( 2x / x = 2 ).
  • Navigating multiple-choice distractors quickly and accurately under timed exam conditions.
Question 3 • Pure Core • 1 Mark

Simplifying log₃ 2x − log₃ x

Multiple-Choice Elimination & Single-Logarithm Form

Given options:

log₃ (1/2) ✔ log₃ 2 log₃ x log₃ 2x²

✅ Correct Answer

The correct option to circle is the second option:

log₃ 2

Mark Scheme Reference:
• B1 (AO 1.1b) awarded for circling the second option ( log₃ 2 ).

📐 Step-by-Step Solution

  1. Identify the base and rule: Both terms have base 3. Use the quotient law:
    log₃ A − log₃ B = log₃ (A / B)
  2. Substitute terms:
    log₃ 2x − log₃ x = log₃ (2x / x)
  3. Simplify the argument: Cancel the common factor of x (where x > 0 ):
    2x / x = 2
  4. Final simplified expression:
    log₃ 2

💡 Key Knowledge: The 3 Core Log Laws

  • Addition Law: logₐ x + logₐ y = logₐ (xy)
  • Subtraction Law: logₐ x − logₐ y = logₐ (x / y)
  • Power Law: k logₐ x = logₐ (xᵏ)
  • Base Matching: These rules only work when the bases are identical (here, both are base 3).

❌ Common Errors & Distractor Traps

  • Treating logs like standard linear terms: Thinking log₃ 2x − log₃ x = log₃ (2x − x) = log₃ x . This is fundamentally incorrect; you divide the arguments, not subtract them!
  • Inverting the quotient: Calculating x / 2x = 1/2 to give log₃ (1/2) . The first argument is always the numerator.
  • Confusing subtraction with addition: Multiplying arguments to give log₃ (2x · x) = log₃ 2x² .

🧠 Exam Technique & Examiner Insight

  • Circle clearly: Multiple choice questions state "Circle your answer". If you change your mind, clearly cross out your previous choice and circle the new one so there is zero ambiguity for the marker.
  • Numerical Substitution Check: If you are ever unsure, substitute a simple number like x = 3 into the initial expression:
    log₃(6) − log₃(3) = log₃(6/3) = log₃(2) . This confirms immediately that the variable x cancels out!
  • Pace yourself: A 1-mark B-grade question like this should take under 30 seconds. Confident mastery of log laws secures rapid marks early in the paper.

Topics

Pure Mathematics · F: Exponentials and logarithms

Question and mark scheme from the AQA A-Level Mathematics examination, Paper 1, June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.