Two unrelated log/algebra parts - linked parts (hence)

Multi-part questions where earlier parts build expressions (e.g., express in terms of y) that are then used via 'hence' in a later part to solve an equation.

9 questions · Moderate -0.7

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OCR C2 2006 January Q7
8 marks Moderate -0.5
7
  1. Express each of the following in terms of \(\log _ { 10 } x\) and \(\log _ { 10 } y\).
    1. \(\log _ { 10 } \left( \frac { x } { y } \right)\)
    2. \(\log _ { 10 } \left( 10 x ^ { 2 } y \right)\)
    3. Given that $$2 \log _ { 10 } \left( \frac { x } { y } \right) = 1 + \log _ { 10 } \left( 10 x ^ { 2 } y \right)$$ find the value of \(y\) correct to 3 decimal places.
OCR MEI AS Paper 2 Specimen Q2
4 marks Moderate -0.8
2
  1. Express \(2 \log _ { 3 } x + \log _ { 3 } a\) as a single logarithm.
  2. Given that \(2 \log _ { 3 } x + \log _ { 3 } a = 2\), express \(x\) in terms of \(a\).
Edexcel C2 Q6
9 marks Moderate -0.8
Given that log₂ x = a, find, in terms of a, the simplest form of
  1. log₂ (16x), [2]
  2. log₂ \(\left(\frac{x⁴}{2}\right)\). [3]
  1. Hence, or otherwise, solve $$\log_2 (16x) - \log_2 \left(\frac{x^4}{2}\right) = \frac{1}{2},$$ giving your answer in its simplest surd form. [4]
Edexcel C2 Q4
9 marks Moderate -0.8
Given that \(\log_2 x = a\), find, in terms of \(a\), the simplest form of
  1. \(\log_2 (16x)\), [2]
  2. \(\log_2 \left( \frac{x^4}{2} \right)\). [3]
  3. Hence, or otherwise, solve \(\log_2 (16x) - \log_2 \left( \frac{x^4}{2} \right) = \frac{1}{2}\), giving your answer in its simplest surd form. [4]
OCR C2 Q4
7 marks Moderate -0.3
  1. Given that \(y = \log_2 x\), find expressions in terms of \(y\) for
    1. \(\log_2 \left(\frac{x}{2}\right)\), [2]
    2. \(\log_2 (\sqrt{x})\). [2]
  2. Hence, or otherwise, solve the equation $$2 \log_2 \left(\frac{x}{2}\right) + \log_2 (\sqrt{x}) = 8.$$ [3]
SPS SPS SM 2021 November Q5
4 marks Moderate -0.3
  1. Write \(\log_{16} y - \log_{16} x\) as a single logarithm. [1]
  2. Solve the simultaneous equations, giving your answers in an exact form. $$\log_3 y = \log_3(9 - 6x) + 1$$ $$\log_{16} y - \log_{16} x = \frac{1}{4}$$ [3]
SPS SPS SM 2022 October Q5
7 marks Moderate -0.8
  1. Given that $$y = \log_3 x$$ find expressions in terms of \(y\) for
    1. \(\log_3\left(\frac{x}{9}\right)\)
    2. \(\log_3 \sqrt{x}\)
    Write each answer in its simplest form. [3]
  2. Hence or otherwise solve $$2\log_3\left(\frac{x}{9}\right) - \log_3 \sqrt{x} = 2$$ [4]
SPS SPS SM 2025 November Q4
9 marks Moderate -0.8
Given that \(\log_2 x = a\), find, in terms of \(a\), the simplest form of
  1. \(\log_2 (16x)\), [2]
  2. \(\log_2 \left(\frac{x^4}{2}\right)\) [3]
  3. Hence, or otherwise, solve $$\log_2 (16x) - \log_2 \left(\frac{x^4}{2}\right) = \frac{1}{2},$$ giving your answer in its simplest surd form. [4]
(Total 9 marks)
Pre-U Pre-U 9794/1 2011 June Q4
6 marks Moderate -0.8
  1. Show that \(4 \ln x - \ln(3x - 2) - \ln x^2 = \ln\left(\frac{x^2}{3x - 2}\right)\), where \(x > \frac{2}{3}\). [3]
  2. Hence solve the equation \(4 \ln x - \ln(3x - 2) - \ln x^2 = 0\). [3]