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0606/12 Feb/Mar 2026, Question 10: Solving Exponential and Log Equations

Cambridge IGCSE Additional Mathematics 0606 · Paper 12 (Non-calculator) · Non-calculator · 10 marks

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The question, in short

Solve two logarithmic and exponential equations: one involving log base 5 and log base x, and one with powers of 2 and 8, finding the values of x.

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

Part (a) · 6 marks

Solve log₅(x²) − log₅ x − 3 log_x 5 = 2 for x.

  1. 1log₅(x²) − log₅ x = 2log₅ x − log₅ x = log₅ x
  2. 2Change of base: log_x 5 = 1/log₅ x
  3. 3Let u = log₅ x: u − 3/u = 2, so u² − 2u − 3 = 0
  4. 4(u − 3)(u + 1) = 0, u = 3 or u = −1
  5. 5x = 5³ = 125 or x = 5⁻¹ = 1/5

Answerx = 125, x = 1/5

Part (b) · 4 marks

Solve 2 × 8^(2x+1) − 3 × 2^(3x+4) − 64 = 0 for x.

  1. 18^(2x+1) = 8 × (2^(3x))² and 2^(3x+4) = 16 × 2^(3x)
  2. 216(2^(3x))² − 48 × 2^(3x) − 64 = 0
  3. 3Let u = 2^(3x): u² − 3u − 4 = 0, so (u − 4)(u + 1) = 0
  4. 42^(3x) = 4 (2^(3x) = −1 has no solution)
  5. 53x = 2, x = 2/3

Answerx = 2/3

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Also searched as: 0606/12/F/M/26 · 0606_m26_qp_12 · 0606 m26 12 · question 10.
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