AQA A-Level Mathematics Paper 3, June 2025: Question 4

3 marks · Easy difficulty · Short Answer

Evaluate logarithmic expressions involving powers of a, including log_a(a^2) and log_a(sqrt(a)) - log_a(1/a^2).

Practise this question

Question

Question 4 contains two parts. Part (a) asks to state the value of log base a of a squared, where a is greater than 0, for 1 mark. Part (b) asks to find the value of log base a of the square root of a minus log base a of 1 over a squared, where a is greater than 0, for 2 marks.

Mark scheme

Show the mark scheme Mark scheme for Question 4. Part 4(a) awards B1 (AO 1.2) for obtaining 2. Part 4(b) awards M1 (AO 1.1a) for correctly using at least one log rule or correctly substituting a value of a > 0 into the full expression leading to a positive value; A1 (AO 1.1b) for obtaining 2.5 (or 5/2). Typical solution shows 1/2 - (-2) = 5/2.

How to answer it

Laws of Logarithms & Index Notation

📌 What this question tests

This question assesses your fluency with the fundamental definitions and laws of logarithms combined with fractional and negative indices:

  • Applying the identity loga(ak) = k (the inverse property of logs).
  • Converting roots into fractional powers: √a = a1/2 .
  • Converting reciprocal terms into negative powers: 1 / a2 = a-2 .
  • Accurate handling of double negatives in logarithmic subtractions.

Question 4 (a)

State the value of loga(a²) where a > 0 [1 mark]

✅ Correct Answer

2

[B1] Awarded for stating 2 directly without needing working.

💡 Key Knowledge

By definition, loga(x) asks: "To what power must base a be raised to equal x?"

  • Power rule: loga(a2) = 2 loga(a)
  • Since loga(a) = 1 , the result simplifies to 2 × 1 = 2 .

❌ Common Errors

  • Leaving the base in the final expression (e.g. writing 2a or a² ).
  • Confusing the base and argument, writing 1/2 .

🧠 Exam Technique

The command word "State" signals that no extensive working is expected. Write the single numerical value clearly on the answer line to secure the B mark immediately.

Question 4 (b)

Find the value of loga(√a) − loga(1 / a²) where a > 0 [2 marks]

✅ Correct Answer

2.5  or  5/2

[M1] Correct use of at least one log/index rule (or valid substitution of a > 0 leading to a positive value).
[A1] Correct final value of 2.5 or 5/2 . Note: Do not ISW (Ignore Subsequent Working).

📐 Step-by-Step Calculation

  1. Rewrite arguments using index notation:
    • √a = a1/2
    • 1 / a2 = a-2
  2. Apply logarithm definition / power rule:
    loga(a1/2) = 1/2
    loga(a-2) = -2 (This earns the M1 mark)
  3. Subtract the terms carefully:
    1/2 − (−2) = 1/2 + 2 = 5/2 = 2.5 (This earns the A1 mark)

🧠 Alternative Method & Tip

Method 2 (Quotient Rule): Combine the arguments first:

loga(√a ÷ 1/a²) = loga(a1/2 × a²) = loga(a5/2) = 5/2

Smart Check: Since the identity must hold for any a > 0 (and a ≠ 1 ), pick an easy test value like a = 4 and evaluate on your calculator to verify your answer!

❌ Common Errors & Examiner Traps

  • Sign Slip on Subtraction: Writing 1/2 − 2 = -1.5 or -3/2 instead of dealing with the double negative 1/2 − (-2) .
  • Fractional Index Mistake: Treating √a as a-1/2 or a2 .
  • "Do Not ISW" Warning: If you find 5/2 correctly but then perform invalid extra algebra (such as multiplying by 2 to get 5), you will lose the final A1 mark! Stop once you reach the evaluated number.

Topics

Pure Mathematics · F: Exponentials and logarithms · B: Algebra and functions

Question and mark scheme from the AQA A-Level Mathematics examination, Paper 3, June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.