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4PM1/02R Jun 2025, Question 9: Solving Exponential and Log Equations

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2R · Calculator · 9 marks

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The question, in short

Solve two logarithmic equations: one for a in the form a = b^c with b prime, and one for p.

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

Part (i) · 3 marks

Solve 3(log_a 9 + log_a 27) = 1, giving a as b^c with b prime and c an integer.

  1. 1log_a 9 + log_a 27 = 2log_a 3 + 3log_a 3 = 5log_a 3
  2. 215log_a 3 = 1, so log_a 3 = 1/15
  3. 3a^(1/15) = 3
  4. 4a = 3¹⁵

Answera = 3¹⁵

Part (ii) · 6 marks

Solve log₄ p + log_p 256 = −4 for p.

  1. 1log_p 256 = log₄ 256/log₄ p = 4/log₄ p
  2. 2log₄ p + 4/log₄ p = −4
  3. 3(log₄ p)² + 4log₄ p + 4 = 0
  4. 4(log₄ p + 2)² = 0, so log₄ p = −2
  5. 5p = 4⁻² = 1/16

Answerp = 1/16

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