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4PM1/01R Jun 2022, Question 7: Solving Exponential and Log Equations

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1R · Calculator · 11 marks

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The question, in short

Sketch the log base 10 curve y = log₁₀(x + 2) on given axes with axis intercepts labelled, then solve two logarithmic equations involving bases a and q.

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

Part (a) · 2 marks

On the provided blank axes, draw the graph of y = log₁₀(x + 2) and label where it crosses the x-axis and y-axis.

  1. 1y = 0: x + 2 = 1 ⇒ x = −1
  2. 2x = 0: y = log₁₀ 2 ≈ 0.301
  3. 3The curve increases and approaches x = −2 as a vertical asymptote

AnswerIncreasing log-shaped curve, asymptotic to x = −2, crossing the x-axis at (−1, 0) and the y-axis at (0, log₁₀ 2) ≈ (0, 0.301)

Part (b) · 3 marks

Solve 2(log_a 4 + log_a 16) = 1 for a.

  1. 1log_a 16 = 2 log_a 4, so 2(3 log_a 4) = 1
  2. 2log_a 4 = 1/6
  3. 3a^(1/6) = 4 ⇒ a = 4⁶ = 4096

Answera = 4096

Part (c) · 6 marks

Solve 5 log_q 16 + 4 log₂ q = 24 for q.

  1. 1log_q 16 = log₂ 16/log₂ q = 4/log₂ q
  2. 25 × 4/log₂ q + 4 log₂ q = 24
  3. 3Let u = log₂ q: 20 + 4u² = 24u ⇒ u² − 6u + 5 = 0
  4. 4(u − 5)(u − 1) = 0 ⇒ u = 5 or u = 1
  5. 5q = 2⁵ = 32 or q = 2¹ = 2

Answerq = 2 or q = 32

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