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WMA12/01 Oct 2022, Question 4: The Laws of Logarithms

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

The question, in short

Given w = logₐ(t + 5) − logₐ4 for 2 ≤ t ≤ 18 and w = 10 when t = 3, work with this logarithmic model of a baby mammal's weight.

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

Part (a) · 3 marks

Use the given weight at t = 3 to show that the constant a equals 1.072 to 3 decimal places.

  1. 1Substitute t = 3, w = 10: 10 = logₐ8 − logₐ4
  2. 2Subtraction law: 10 = logₐ(8/4) = logₐ2
  3. 3So a¹⁰ = 2
  4. 4a = 2^(1/10) = 1.07177… = 1.072 to 3 d.p.

AnswerShown: logₐ2 = 10 ⇒ a¹⁰ = 2 ⇒ a = 2^(1/10) = 1.07177… = 1.072 (3 d.p.)

Part (b) · 3 marks

Using a = 1.072, rearrange the logarithmic formula to express t in terms of w.

  1. 1w = log₁.₀₇₂((t + 5)/4)
  2. 2(t + 5)/4 = 1.072ʷ
  3. 3t = 4 × 1.072ʷ − 5

Answert = 4 × 1.072ʷ − 5

Part (c) · 2 marks

Using a = 1.072, calculate t when w = 15, giving the answer to 3 significant figures.

  1. 1t = 4 × 1.072¹⁵ − 5
  2. 21.072¹⁵ = 2.8374…
  3. 3t = 11.349… − 5 = 6.35 (3 s.f.)

Answert = 6.35

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