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WMA12/01 Jan 2024, Question 6: The Laws of Logarithms

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 8 marks

The question, in short

Using log laws, show a given log equation leads to the cubic x³ + 6x² + x − 4 = 0, then find its other roots and solve the original log equation.

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

Part (a) · 4 marks

Combine the logarithms and the constant 1/2 to derive the cubic equation x³ + 6x² + x − 4 = 0.

  1. 12log₄(x + 3) = log₄(x + 3)² and 1/2 = log₄2
  2. 2log₄[x(x + 3)²] = log₄(4x + 2) + log₄2 = log₄[2(4x + 2)]
  3. 3x(x + 3)² = 8x + 4
  4. 4x(x² + 6x + 9) = 8x + 4
  5. 5x³ + 6x² + 9x − 8x − 4 = 0, so x³ + 6x² + x − 4 = 0

AnswerShown: x(x + 3)² = 2(4x + 2) ⇒ x³ + 6x² + 9x = 8x + 4 ⇒ x³ + 6x² + x − 4 = 0

Part (bi) · 3 marks

Given that −1 is a root, use algebra to find the remaining two roots of the cubic.

  1. 1x³ + 6x² + x − 4 = (x + 1)(x² + 5x − 4)
  2. 2Solve x² + 5x − 4 = 0
  3. 3x = (−5 ± √(25 + 16))/2 = (−5 ± √41)/2

Answerx = (−5 + √41)/2 and x = (−5 − √41)/2

Part (bii) · 1 mark

Use the roots found to solve the original logarithmic equation, checking which values are valid.

  1. 1log₄x is only defined for x > 0
  2. 2−1 and (−5 − √41)/2 are negative, so reject them
  3. 3x = (−5 + √41)/2 ≈ 0.702

Answerx = (−5 + √41)/2

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