Express log in terms of given variables

Given that certain logarithms equal variables (e.g., log_a(b)=p), express another logarithm in terms of those variables.

15 questions · Moderate -0.7

1.06f Laws of logarithms: addition, subtraction, power rules
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CAIE P3 2024 March Q4
4 marks Moderate -0.5
4 The positive numbers \(p\) and \(q\) are such that $$\ln \left( \frac { p } { q } \right) = a \text { and } \ln \left( q ^ { 2 } p \right) = b .$$ Express \(\ln \left( p ^ { 7 } q \right)\) in terms of \(a\) and \(b\).
Edexcel C12 2019 June Q13
6 marks Moderate -0.8
13. Given that \(p = \log _ { a } 9\) and \(q = \log _ { a } 10\), where \(a\) is a constant, find in terms of \(p\) and \(q\),
  1. \(\log _ { a } 900\)
  2. \(\log _ { a } 0.3\)
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Edexcel P2 2019 October Q7
7 marks Moderate -0.3
  1. Given \(\log _ { a } b = k\), find, in simplest form in terms of \(k\),
    1. \(\log _ { a } \left( \frac { \sqrt { a } } { b } \right)\)
    2. \(\frac { \log _ { a } a ^ { 2 } b } { \log _ { a } b ^ { 3 } }\)
    3. \(\sum _ { n = 1 } ^ { 50 } \left( k + \log _ { a } b ^ { n } \right)\)
Edexcel P2 2022 October Q10
7 marks Moderate -0.3
  1. Given \(a = \log _ { 2 } 3\)
    1. write, in simplest form, in terms of \(a\),
      (a) \(\log _ { 2 } 9\) (b) \(\log _ { 2 } \left( \frac { \sqrt { 3 } } { 16 } \right)\)
    2. Solve
    $$3 ^ { x } \times 2 ^ { x + 4 } = 6$$ giving your answer, in simplest form, in terms of \(a\).
Edexcel C2 2013 June Q6
9 marks Moderate -0.3
6. Given that \(\log _ { 3 } x = a\), find in terms of \(a\),
  1. \(\log _ { 3 } ( 9 x )\)
  2. \(\log _ { 3 } \left( \frac { x ^ { 5 } } { 81 } \right)\) giving each answer in its simplest form.
  3. Solve, for \(x\), $$\log _ { 3 } ( 9 x ) + \log _ { 3 } \left( \frac { x ^ { 5 } } { 81 } \right) = 3$$ giving your answer to 4 significant figures. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{4f4eac7b-8908-480f-bb39-049944203fff-10_775_1605_221_159} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} The line with equation \(y = 10\) cuts the curve with equation \(y = x ^ { 2 } + 2 x + 2\) at the points \(A\) and \(B\) as shown in Figure 1. The figure is not drawn to scale.
OCR C2 2009 January Q8
10 marks Moderate -0.8
8
  1. Given that \(\log _ { a } x = p\) and \(\log _ { a } y = q\), express the following in terms of \(p\) and \(q\).
    1. \(\log _ { a } ( x y )\)
    2. \(\log _ { a } \left( \frac { a ^ { 2 } x ^ { 3 } } { y } \right)\)
    1. Express \(\log _ { 10 } \left( x ^ { 2 } - 10 \right) - \log _ { 10 } x\) as a single logarithm.
    2. Hence solve the equation \(\log _ { 10 } \left( x ^ { 2 } - 10 \right) - \log _ { 10 } x = 2 \log _ { 10 } 3\).
Edexcel AS Paper 1 2024 June Q9
5 marks Moderate -0.3
9. $$\begin{aligned} p & = \log _ { a } 16 \\ q & = \log _ { a } 25 \end{aligned}$$ where \(a\) is a constant.
Find in terms of \(p\) and/or \(q\),
  1. \(\log _ { a } 256\)
  2. \(\log _ { a } 100\)
  3. \(\log _ { a } 80 \times \log _ { a } 3.2\)
Edexcel Paper 1 2023 June Q6
6 marks Moderate -0.3
6. $$a = \log _ { 2 } x \quad b = \log _ { 2 } ( x + 8 )$$ Express in terms of \(a\) and/or \(b\)
  1. \(\log _ { 2 } \sqrt { x }\)
  2. \(\log _ { 2 } \left( x ^ { 2 } + 8 x \right)\)
  3. \(\log _ { 2 } \left( 8 + \frac { 64 } { x } \right)\) Give your answer in simplest form.
Edexcel C2 Q1
6 marks Moderate -0.8
  1. Given that \(p = \log _ { q } 16\), express in terms of \(p\),
    1. \(\log _ { q } 2\),
    2. \(\log _ { q } ( 8 q )\).
      [0pt] [P2 January 2002 Question 2]
    3. \(\mathrm { f } ( x ) = x ^ { 3 } - x ^ { 2 } - 7 x + c\), where \(c\) is a constant.
    Given that \(\mathrm { f } ( 4 ) = 0\),
  2. find the value of \(c\),
  3. factorise \(\mathrm { f } ( x )\) as the product of a linear factor and a quadratic factor.
  4. Hence show that, apart from \(x = 4\), there are no real values of \(x\) for which \(\mathrm { f } ( x ) = 0\).
Edexcel C2 Q3
6 marks Moderate -0.8
3. Given that \(p = \log _ { 2 } 3\) and \(q = \log _ { 2 } 5\), find expressions in terms of \(p\) and \(q\) for
  1. \(\quad \log _ { 2 } 45\),
  2. \(\quad \log _ { 2 } 0.3\)
OCR C2 Q2
6 marks Moderate -0.8
2. Given that \(p = \log _ { 2 } 3\) and \(q = \log _ { 2 } 5\), find expressions in terms of \(p\) and \(q\) for
  1. \(\quad \log _ { 2 } 45\),
  2. \(\log _ { 2 } 0.3\)
Edexcel C2 Q18
6 marks Moderate -0.8
Given that \(p = \log_4 16\), express in terms of \(p\),
  1. \(\log_4 2\). [2]
  2. \(\log_4 (8q)\). [4]
Edexcel C2 Q1
6 marks Moderate -0.8
Given that \(p = \log_q 16\), express in terms of \(p\),
  1. \(\log_q 2\). [2]
  2. \(\log_q (8q)\). [4]
SPS SPS FM 2020 October Q2
3 marks Easy -1.2
Let \(a = \log_2 x\), \(b = \log_2 y\) and \(c = \log_2 z\). Express \(\log_2(xy) - \log_2(\frac{z}{y})\) in terms of \(a\), \(b\) and \(c\). [3]
SPS SPS SM 2020 October Q5
3 marks Easy -1.2
Let \(a = \log_2 x\), \(b = \log_2 y\) and \(c = \log_2 z\). Express \(\log_2(xy) - \log_2(\frac{z^2}{x})\) in terms of \(a\), \(b\) and \(c\). [3]