Indices and logarithms — A/L Combined Mathematics

How do I practise Indices and logarithms?

1 question with a worked solution

1. What is the real solution to the logarithmic equation log₂(x) + log₂(x - 2) = 3?
   1. A) x = 4 and x = -2
   2. B) x = 4
   3. C) x = 8
   4. D) x = -2
   Check my answer

   Correct answer: B) x = 4

   Using log_b(u) + log_b(v) = log_b(uv): log₂[x(x - 2)] = 3 => x(x - 2) = 2³ = 8 => x² - 2x - 8 = 0 => (x - 4)(x + 2) = 0. Since the argument of a logarithm must be strictly positive (x > 0 and x - 2 > 0 => x > 2), the only valid real root is x = 4 (x = -2 is rejected).

   Show worked solution

   Step 1: Combine logs: log₂[x(x - 2)] = 3 => x² - 2x = 8. Step 2: Factor: (x - 4)(x + 2) = 0 => x = 4 or x = -2. Step 3: Check domain: x > 2, so reject x = -2. Valid root: x = 4. Answer: x = 4.
   Modelled on: the A/L Combined Mathematics examination pattern

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Where does Indices and logarithms fit in the syllabus?

Year 1 · Term 1