Two unrelated log/algebra parts - simplify/express then solve

Two-part questions where one part asks to simplify, express, or rewrite a logarithmic expression, and the other part asks to solve an equation or find a value, with no substitution link between them.

8 questions · Moderate -0.8

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Edexcel P2 2024 June Q3
6 marks Moderate -0.3
  1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
    1. Using the laws of logarithms, solve
    $$2 \log _ { 2 } ( 2 - x ) = 4 + \log _ { 2 } ( x + 10 )$$
  2. Find the value of $$\log _ { \sqrt { a } } a ^ { 6 }$$ where \(a\) is a positive constant greater than 1
AQA C2 2008 January Q7
4 marks Easy -1.2
7
  1. Given that $$\log _ { a } x = \log _ { a } 16 - \log _ { a } 2$$ write down the value of \(x\).
  2. Given that $$\log _ { a } y = 2 \log _ { a } 3 + \log _ { a } 4 + 1$$ express \(y\) in terms of \(a\), giving your answer in a form not involving logarithms.
AQA C2 2006 June Q5
6 marks Moderate -0.8
5
  1. Given that $$\log _ { a } x = 2 \log _ { a } 6 - \log _ { a } 3$$ show that \(x = 12\).
  2. Given that $$\log _ { a } y + \log _ { a } 5 = 7$$ express \(y\) in terms of \(a\), giving your answer in a form not involving logarithms.
    (3 marks)
AQA C2 2007 June Q8
8 marks Moderate -0.8
8
  1. It is given that \(n\) satisfies the equation $$\log _ { a } n = \log _ { a } 3 + \log _ { a } ( 2 n - 1 )$$ Find the value of \(n\).
  2. Given that \(\log _ { a } x = 3\) and \(\log _ { a } y - 3 \log _ { a } 2 = 4\) :
    1. express \(x\) in terms of \(a\);
    2. express \(x y\) in terms of \(a\).
OCR C2 Q7
7 marks Moderate -0.8
  1. Evaluate \(\log_3 15 + \log_3 20 - \log_3 12\). [3]
  2. Given that \(y = 3 \times 10^{2x}\), show that \(x = a \log_{10}(by)\), where the values of the constants \(a\) and \(b\) are to be found. [4]
OCR C2 Specimen Q3
7 marks Moderate -0.8
  1. Express each of the following in terms of \(\log_2 x\):
    1. \(\log_2(x^2)\), [1]
    2. \(\log_2(8x^2)\). [3]
  2. Given that \(y^2 = 27\), find the value of \(\log_3 y\). [3]
OCR MEI C2 2016 June Q8
5 marks Moderate -0.8
  1. Simplify \(\log_a 1 - \log_a (a^m)^3\). [2]
  2. Use logarithms to solve the equation \(3^{2x+1} = 1000\). Give your answer correct to 3 significant figures. [3]
AQA Paper 3 2018 June Q7
5 marks Moderate -0.8
  1. Given that \(\log_a y = 2\log_a 7 + \log_a 4 + \frac{1}{2}\), find \(y\) in terms of \(a\). [4 marks]
  2. When asked to solve the equation $$2\log_a x = \log_a 9 - \log_a 4$$ a student gives the following solution: \(2\log_a x = \log_a 9 - \log_a 4\) \(\Rightarrow 2\log_a x = \log_a \frac{9}{4}\) \(\Rightarrow \log_a x^2 = \log_a \frac{9}{4}\) \(\Rightarrow x^2 = \frac{9}{4}\) \(\therefore x = \frac{3}{2}\) or \(-\frac{3}{2}\) Explain what is wrong with the student's solution. [1 mark]