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4PM1/01 Jun 2021, Question 8: The Laws of Logarithms

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 8 marks

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The question, in short

Solve a logarithmic equation in base a for n, then use given log equations in base p to express x and the product xy in terms of p.

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

Part (a) · 3 marks

Use log laws to combine the right side, equate arguments, and solve the resulting equation for n, checking validity.

  1. 1log_a n = log_a 3(2n − 1)
  2. 2n = 3(2n − 1) = 6n − 3
  3. 35n = 3, n = 3/5

Answern = 3/5

Part (bi) · 1 mark

Rewrite the equation log_p x = 3 in exponential form to give x in terms of p.

  1. 1log_p x = 3 ⇒ x = p³

Answerx = p³

Part (bii) · 4 marks

Combine the second log equation using log laws, then use x from part (i) to express xy in terms of p.

  1. 13 log_p 2 = log_p 8
  2. 2log_p y − log_p 8 = 4 ⇒ log_p (y/8) = 4
  3. 3y/8 = p⁴, so y = 8p⁴
  4. 4xy = p³ × 8p⁴ = 8p⁷

Answerxy = 8p⁷

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