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0606/13 May/Jun 2026, Question 3: Solving Exponential and Log Equations

Cambridge IGCSE Additional Mathematics 0606 · Paper 13 (Non-calculator) · Non-calculator · 12 marks

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The question, in short

Solve three separate equations: a logarithmic equation in base x, an exponential equation in 2 to the power x, and a modulus quadratic equation.

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

Part (a) · 4 marks

Solve log base x of 9 equals log base 9 of x to the fourth power.

  1. 1log_x 9 = 1/log₉ x and log₉(x⁴) = 4 log₉ x
  2. 21/log₉ x = 4 log₉ x
  3. 3(log₉ x)² = 1/4
  4. 4log₉ x = ±1/2
  5. 5x = 9^(1/2) = 3 or x = 9^(−1/2) = 1/3

Answerx = 3, x = 1/3

Part (b) · 4 marks

Solve 2 to the power 2x−2 plus 3 times 2 to the power x minus 40 equals 0.

  1. 12^(2x−2) = (1/4)(2^(2x)), so (1/4)(2^(2x)) + 3(2ˣ) − 40 = 0
  2. 22^(2x) + 12(2ˣ) − 160 = 0
  3. 3(2ˣ + 20)(2ˣ − 8) = 0
  4. 42ˣ = 8 (2ˣ = −20 has no solution)
  5. 5x = 3

Answerx = 3

Part (c) · 4 marks

Solve the modulus of x squared plus 3x minus 4 equals 6.

  1. 1x² + 3x − 4 = 6 or x² + 3x − 4 = −6
  2. 2x² + 3x − 10 = 0: (x − 2)(x + 5) = 0, x = 2, −5
  3. 3x² + 3x + 2 = 0: (x + 1)(x + 2) = 0, x = −1, −2

Answerx = 2, −5, −1, −2

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Also searched as: 0606/13/M/J/26 · 0606_s26_qp_13 · 0606 s26 13 · question 3.
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