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WMA12/01 Jan 2025, Question 7: The Laws of Logarithms

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 8 marks

The question, in short

Apply the trapezium rule to tabulated values of log₁₀(x + 5) to express the integral as log₁₀ k, then solve a logarithmic equation in base 5 for a.

Summarised in our own words. Use the official paper for the exact wording and any diagrams.

Part (i) · 3 marks

Use the trapezium rule with all six y values from x = −1 to 4 to show the integral equals log₁₀ k, finding integer k.

  1. 1h = 1
  2. 2Area ≈ ½ × 1 × [log 4 + log 9 + 2(log 5 + log 6 + log 7 + log 8)]
  3. 3= ½ [log(4 × 9) + 2 log(5 × 6 × 7 × 8)] = ½ log 36 + log 1680
  4. 4= log 6 + log 1680 = log₁₀ 10080, so k = 10080

AnswerShown: ∫ ≈ log₁₀ 10080, so k = 10080

Part (ii) · 5 marks

Solve 2 log₅(5 − a) − log₅(a + 25) = 1 for the value of a.

  1. 12 log₅(5 − a) = log₅(5 − a)²
  2. 2log₅[(5 − a)²/(a + 25)] = 1
  3. 3(5 − a)² = 5(a + 25)
  4. 425 − 10a + a² = 5a + 125, so a² − 15a − 100 = 0
  5. 5(a − 20)(a + 5) = 0
  6. 6a = 20 makes 5 − a negative, so reject it: a = −5

Answera = −5

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