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4PM1/01R Jan 2020, Question 5: Solving Exponential and Log Equations

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1R · Calculator · 9 marks

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The question, in short

Show a logarithmic value equals 5/2, then use it to solve an equation involving log base 2, log base 4 and log base x terms exactly.

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

Part (a) · 2 marks

Show that log base 4 of 32 equals 5/2.

  1. 1log₄ 32 = log₂ 32 / log₂ 4
  2. 2= 5/2

Answerlog₄ 32 = 5/2 shown

Part (b) · 7 marks

Using the result or otherwise, find the exact solutions of log₂ x − log₄ 32 + (1/4) logₓ 16 = 0.

  1. 1Let y = log₂ x. logₓ 16 = log₂ 16 / log₂ x = 4/y
  2. 2y − 5/2 + (1/4)(4/y) = 0, so y − 5/2 + 1/y = 0
  3. 32y² − 5y + 2 = 0
  4. 4(2y − 1)(y − 2) = 0, so y = 1/2 or y = 2
  5. 5x = 2^(1/2) = √2 or x = 2² = 4

Answerx = √2, x = 4

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