AQA AS Level Mathematics Paper 1, June 2025: Question 2

1 mark · Easy difficulty · Short Answer

Find the value of $\log_b \frac{1}{b^2}$ from multiple choice options.

Practise this question

Question

Question 2 asks: 'Find the value of log base b of 1 over b squared. Circle your answer.' Below the text are four horizontal options: -2, -1/2, 1/2, and 2. It is worth 1 mark.
Question text

2 Find the value of logb 2

b

Circle your answer.

[1 mark]

–2 – 2

Mark scheme

Show the mark scheme Mark scheme table for Question 2 with AO 1.1b, award B1 for 'Circles first answer', with typical solution given as -2.

Q Marking instructions AO Marks Typical solution

2 Circles first answer 1.1b B1 –2

Question 2 Total 1

How to answer it

Evaluating Logarithms with Negative Indices

What this question tests

This multiple-choice question assesses your fundamental understanding of logarithmic definitions, the power law of logarithms, and negative index laws (converting reciprocal algebraic powers to standard index form).

Question 2

Multiple Choice [1 Mark]

Find the value of logb(1 / b²) . Circle your answer.

−2  ✓
−1/2
1/2
2
Mark Scheme Allocation:
• B1 (AO 1.1b): Correctly circles first answer ( −2 ).

Step-by-Step Solutions

📐 Method 1: Negative Exponents & Power Law

  1. Rewrite the reciprocal using negative indices:
    Recall that 1 / bⁿ = b⁻ⁿ.
    Therefore, 1 / b² = b⁻² .
  2. Substitute into the log expression:
    logb(1 / b²) = logb(b⁻²)
  3. Apply the power law of logarithms:
    Using loga(xᵏ) = k loga(x) :
    logb(b⁻²) = −2 logb(b)
  4. Evaluate log to its base:
    Since logb(b) = 1 :
    −2 × 1 = −2

📐 Method 2: Equating to an Unknown (Definition)

  1. Set the expression equal to x:
    Let x = logb(1 / b²) .
  2. Convert from logarithmic to exponential form:
    By definition, loga(y) = x ⇔ aˣ = y .
    So, bˣ = 1 / b² .
  3. Express both sides as powers of b:
    bˣ = b⁻²
  4. Equate the indices:
    Comparing powers of equal bases gives x = −2 .

✅ Correct Answer

The correct option to circle is −2.

Only one option should be clearly circled or indicated. If you change your mind, cross out the incorrect choice cleanly.

💡 Key Knowledge

  • Reciprocal Index Law: 1 / xⁿ = x⁻ⁿ
  • Base Identity: loga(a) = 1 for any base a > 0, a ≠ 1
  • Log Power Law: loga(xᵏ) = k loga(x)
  • Log Quotient Law (alternative): logb(1 / b²) = logb(1) − logb(b²) = 0 − 2 = −2

❌ Common Errors & Traps

  • Confusing reciprocals with fractional powers: Mistaking 1 / b² for b^(1/2) or b^(−1/2) , leading students to circle 1/2 or −1/2 .
  • Ignoring the fraction: Evaluating logb(b²) = 2 and forgetting the negative sign caused by the reciprocal in the denominator.
  • Ambiguous circling: Circling two answers without striking one out forfeits the B1 mark automatically.

🧠 Examiner Technique

  • Time efficiency: This is a standard 1-mark recall question designed to take under 60 seconds. Do not overcomplicate it.
  • Eliminate distractors: The options include 2 , −2 , 1/2 , and −1/2 . Because 1 / b² < 1 (for b > 1 ), the result must be negative, eliminating positive options immediately.
  • Spotting the distractor logic:
    • 2 comes from ignoring the reciprocal.
    • ±1/2 comes from confusing indices with roots.

Topics

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

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