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WMA12/01 Oct 2023, Question 5: The Laws of Logarithms

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 6 marks

The question, in short

Solve two equations: an exponential equation for a to 3 decimal places, and a logarithmic equation for the exact value of b, showing all working without relying on calculator technology.

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

Part (i) · 2 marks

Solve 3ᵃ = 70, giving the answer to 3 decimal places.

  1. 1Take logs of both sides: a log 3 = log 70
  2. 2a = log 70 ÷ log 3
  3. 3a = 3.867 (3 d.p.)

Answera = 3.867

Part (ii) · 4 marks

Find the exact value of b satisfying 4 + 3 log₃ b = log₃ 5b.

  1. 1Write 4 = log₃ 81 and 3 log₃ b = log₃ b³
  2. 2log₃ 81 + log₃ b³ = log₃ 5b, so log₃(81b³) = log₃(5b)
  3. 381b³ = 5b, and b > 0, so 81b² = 5
  4. 4b² = 5/81, so b = √5/9 (reject the negative root)

Answerb = √(5/81) = √5/9

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