AQA AS Level Mathematics Paper 1, June 2025: Question 6

📐 Step-by-Step Calculation

Step 1: Apply the quotient rule for logarithms:

log₂(x² / y) = log₂(x²) − log₂y

Step 2: Apply the power rule to bring the exponent 2 to the front:

log₂(x²) = 2 log₂x

Giving: 2 log₂x − log₂y

Step 3: Substitute p = log₂x and q = log₂y :

2p − q

✅ Correct Answer

2p − q

Mark Breakdown:

• M1: Correctly uses at least one law of logs (e.g. log₂(x²) − log₂y or 2 log₂(x / √y)).
• A1: Correct final expression 2p − q with no incorrect log laws seen anywhere in the working.

🧠 Exam Technique

• Show all intermediary steps: Accuracy marks (A marks) require that no invalid logarithmic steps appear. Writing out each rule explicitly protects your method mark.
• Check your variables: Ensure you replace every logarithm term with p and q ; the question asks for the answer in terms of p and q.

❌ Common Errors

• Distributing logs incorrectly: Writing log(x² / y) = log(x²) / log(y) is a fatal algebraic error that scores zero marks.
• Squaring the variable: Writing p² − q instead of 2p − q . Remember: log₂(x²) = 2 log₂x = 2p, whereas (log₂x)² = p².

📐 Step-by-Step Calculation

Step 1: Convert the root to a fractional index:

√y = y^1/2

So the argument is: 16 · x³ · y^1/2

Step 2: Split into three separate terms using the product rule:

log₂(16) + log₂(x³) + log₂(y^1/2)

Step 3: Apply the power rule to indices and evaluate the numerical term:

• log₂(16) = 4 (since 2⁴ = 16)
• log₂(x³) = 3 log₂x = 3p
• log₂(y^1/2) = (1/2) log₂y = (1/2)q

Step 4: Combine all evaluated terms:

4 + 3p + (1/2)q

✅ Correct Answer

4 + 3p + ½q

(Equivalent forms like 4 + 3p + 0.5q or 4 + 3p + q/2 are fully accepted)

Mark Breakdown:

• B1: Correctly rewrites √y as y^1/2, implied by seeing (1/2)log₂y or (1/2)q.
• M1: Correctly splits the expression into three separate log terms and applies laws to get three terms.
• A1: Fully simplified expression 4 + 3p + ½q with no logarithms remaining.

💡 Key Knowledge

• Numerical Logs: Always check if a number inside log_b is an integer power of the base b. Here, 16 = 2⁴, so log₂(16) simplifies to a pure constant, 4.
• Non-logarithmic form: The question specifically reminds candidates: "in a form not involving logarithms". Leaving terms like log₂16 forfeits the final accuracy mark.

❌ Common Errors

• Forgetting to evaluate log₂(16): Leaving log₂16 + 3p + ½q or evaluating it incorrectly as 8 (confusing 16 / 2 with 2⁴).
• Multiplication inside vs outside: Writing 4 × 3p × ½q instead of adding the terms together.
• Cube error: Misinterpreting x³ as giving p³ instead of bringing the power to the front as 3p .