Combine logs into single logarithm

Express a sum, difference, or multiple of logarithms as a single logarithm (e.g., log_a(2) + log_a(3) = log_a(6), or 2log_a(6) - log_a(3) as single log).

10 questions · Easy -1.3

1.06f Laws of logarithms: addition, subtraction, power rules
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Edexcel C2 2006 June Q3
4 marks Easy -1.2
  1. Write down the value of \(\log _ { 6 } 36\).
  2. Express \(2 \log _ { a } 3 + \log _ { a } 11\) as a single logarithm to base \(a\).
OCR C2 2008 January Q3
4 marks Easy -1.8
3 Express each of the following as a single logarithm:
  1. \(\log _ { a } 2 + \log _ { a } 3\),
  2. \(2 \log _ { 10 } x - 3 \log _ { 10 } y\).
OCR MEI C2 Q3
4 marks Moderate -0.8
3
  1. Write \(\log _ { 10 } ( x + 4 ) - 2 \log _ { 10 } x + \log _ { 10 } ( x + 16 )\) as a single logarithm.
  2. Without using your calculator, verify that \(x = 4\) is a root of the equation $$\log _ { 10 } ( x + 4 ) - 2 \log _ { 10 } x + \log _ { 10 } ( x + 16 ) = 1$$
AQA C2 2009 January Q6
9 marks Moderate -0.8
6
  1. Write each of the following in the form \(\log _ { a } k\), where \(k\) is an integer:
    1. \(\log _ { a } 4 + \log _ { a } 10\);
    2. \(\log _ { a } 16 - \log _ { a } 2\);
    3. \(3 \log _ { a } 5\).
  2. Use logarithms to solve the equation \(( 1.5 ) ^ { 3 x } = 7.5\), giving your value of \(x\) to three decimal places.
  3. Given that \(\log _ { 2 } p = m\) and \(\log _ { 8 } q = n\), express \(p q\) in the form \(2 ^ { y }\), where \(y\) is an expression in \(m\) and \(n\).
Pre-U Pre-U 9794/2 2012 Specimen Q1
7 marks Easy -1.8
1
  1. Express each of the following as a single logarithm.
    1. \(\log _ { a } 5 + \log _ { a } 3\)
    2. \(5 \log _ { b } 2 - 3 \log _ { b } 4\)
  2. Express \(\left( 9 a ^ { 4 } \right) ^ { - \frac { 1 } { 2 } }\) as an algebraic fraction in its simplest form.
Pre-U Pre-U 9794/2 2016 Specimen Q1
9 marks Easy -1.3
1
  1. Express each of the following as a single logarithm.
    1. \(\log _ { a } 5 + \log _ { a } 3\)
    2. \(5 \log _ { b } 2 - 3 \log _ { b } 4\)
  2. Express \(\left( 9 a ^ { 4 } \right) ^ { - \frac { 1 } { 2 } }\) as an algebraic fraction in its simplest form.
  3. Show that \(\frac { 3 \sqrt { 3 } - 1 } { 2 \sqrt { 3 } - 3 } = \frac { 15 + 7 \sqrt { 3 } } { 3 }\).
Pre-U Pre-U 9794/1 2017 June Q2
5 marks Easy -1.3
2 Express each of the following as a single logarithm.
  1. \(\log 3 + \log 4 - \log 2\)
  2. \(2 \log x - 3 \log y + 2 \log z\)
OCR MEI C2 2010 June Q7
2 marks Easy -1.2
Express \(\log_a x^3 + \log_a \sqrt{x}\) in the form \(k \log_a x\). [2]
AQA AS Paper 1 2022 June Q1
1 marks Easy -1.8
Express as a single logarithm $$\log_{10} 2 - \log_{10} x$$ Circle your answer. [1 mark] \(\log_{10} (2 + x)\) \quad \(\log_{10} (2 - x)\) \quad \(\log_{10} (2x)\) \quad \(\log_{10} \left(\frac{2}{x}\right)\)
AQA AS Paper 2 2018 June Q3
2 marks Easy -1.2
Express as a single logarithm \(2\log_a 6 - \log_a 3\) [2 marks]