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4PM1/01 Jun 2019, Question 9: Solving Exponential and Log Equations

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 12 marks

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The question, in short

Solve a logarithmic equation in p, find k from a log base comparison, and prove a logarithmic identity in x.

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

Part (a) · 6 marks

Find the value of p satisfying 2 log_p 9 + 3 log₃ p = 8.

  1. 1log_p 9 = log₃ 9/log₃ p = 2/log₃ p
  2. 24/log₃ p + 3 log₃ p = 8
  3. 33(log₃ p)² − 8 log₃ p + 4 = 0
  4. 4(3 log₃ p − 2)(log₃ p − 2) = 0
  5. 5log₃ p = 2/3 → p = 3^(2/3) = ∛9 ≈ 2.08
  6. 6log₃ p = 2 → p = 9

Answerp = 9, p = ∛9 (2.08)

Part (b) · 2 marks

Given log₂ 3 = log₄ 3^k, determine k.

  1. 1log₂ 3 = log₄ 3/log₄ 2 = log₄ 3/(1/2)
  2. 2= 2 log₄ 3 = log₄ 3²
  3. 3k = 2

Answerk = 2

Part (c) · 4 marks

Prove the given logarithmic expression equals log₄ of (x^(6x − 5)) ÷ (3^(6x − 20)).

  1. 1log₂ 3 = log₄ 9, so the expression = 6x log₄ x − 5 log₄ x − 3x log₄ 9 + 10 log₄ 9
  2. 2= log₄ x^(6x) − log₄ x⁵ − log₄ 3^(6x) + log₄ 3^(20)
  3. 3= log₄ (x^(6x) × 3^(20))/(x⁵ × 3^(6x))
  4. 4= log₄ (x^(6x − 5)/3^(6x − 20))

Answer6x log₄ x − 3x log₂ 3 − 5 log₄ x + 10 log₂ 3 = log₄(x^(6x − 5)/3^(6x − 20)) shown

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