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0606/12 Oct/Nov 2025, Question 5: Solving Exponential and Log Equations

Cambridge IGCSE Additional Mathematics 0606 · Paper 12 (Non-calculator) · Non-calculator · 7 marks

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The question, in short

Combine a logarithmic expression with coefficients 5, 4 and constant 3 into one base 10 log, then solve an equation involving log base 5 and log base (x+1).

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

Part (a) · 3 marks

Express 5 lg a − 4 lg b − 3 as a single logarithm to base 10.

  1. 15 lg a = lg a⁵ and 4 lg b = lg b⁴
  2. 23 = lg 1000
  3. 3lg a⁵ − lg b⁴ − lg 1000 = lg(a⁵/(1000b⁴))

Answerlg(a⁵/(1000b⁴))

Part (b) · 4 marks

Find x satisfying log base 5 of (x + 1) minus log base (x + 1) of 5 equals 0.

  1. 1Change of base: log_(x+1) 5 = 1/log₅(x + 1)
  2. 2log₅(x + 1) − 1/log₅(x + 1) = 0
  3. 3(log₅(x + 1))² = 1 ⇒ log₅(x + 1) = ±1
  4. 4x + 1 = 5 ⇒ x = 4
  5. 5x + 1 = 1/5 ⇒ x = −4/5

Answerx = 4, x = −4/5

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Also searched as: 0606/12/O/N/25 · 0606_w25_qp_12 · 0606 w25 12 · question 5.
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