Solve ln equation using subtraction law

Equation involves ln(A) - ln(B) = constant or ln expression, solved by combining into ln(A/B) and exponentiating.

17 questions · Moderate -0.4

1.06f Laws of logarithms: addition, subtraction, power rules1.06g Equations with exponentials: solve a^x = b
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CAIE P2 2020 June Q1
3 marks Moderate -0.5
1 Solve the equation $$\ln ( x + 1 ) - \ln x = 2 \ln 2$$
CAIE P2 2020 November Q1
4 marks Moderate -0.5
1 Given that $$\ln ( 2 x + 1 ) - \ln ( x - 3 ) = 2$$ find \(x\) in terms of e.
CAIE P3 2014 June Q1
3 marks Moderate -0.5
1 Solve the equation \(\log _ { 10 } ( x + 9 ) = 2 + \log _ { 10 } x\).
CAIE P3 2008 November Q1
3 marks Moderate -0.5
1 Solve the equation $$\ln ( x + 2 ) = 2 + \ln x$$ giving your answer correct to 3 decimal places.
CAIE P3 2009 November Q1
4 marks Moderate -0.5
1 Solve the equation $$\ln ( 5 - x ) = \ln 5 - \ln x$$ giving your answers correct to 3 significant figures.
CAIE P3 2012 November Q1
3 marks Moderate -0.5
1 Solve the equation $$\ln ( x + 5 ) = 1 + \ln x$$ giving your answer in terms of e.
CAIE P2 2017 November Q1
4 marks Moderate -0.3
1 Solve the equation \(\ln ( 3 x + 1 ) - \ln ( x + 2 ) = 1\), giving your answer in terms of e.
CAIE P3 2023 June Q1
4 marks Moderate -0.5
1 Solve the equation \(\ln ( x + 5 ) = 5 + \ln x\). Give your answer correct to 3 decimal places.
CAIE P3 2024 June Q2
4 marks Standard +0.3
2 Solve the equation \(\ln ( x - 5 ) = 7 - \ln x\). Give your answer correct to 2 decimal places.
OCR C2 Q1
4 marks Moderate -0.3
  1. Solve the equation
$$\log _ { 5 } ( 4 x + 3 ) - \log _ { 5 } ( x - 1 ) = 2$$
OCR C2 2016 June Q4
6 marks Moderate -0.8
4
  1. Express \(2 \log _ { 3 } x - \log _ { 3 } ( x + 4 )\) as a single logarithm.
  2. Hence solve the equation \(2 \log _ { 3 } x - \log _ { 3 } ( x + 4 ) = 2\).
Edexcel Paper 1 2019 June Q9
5 marks Standard +0.3
  1. Given that \(a > b > 0\) and that \(a\) and \(b\) satisfy the equation
$$\log a - \log b = \log ( a - b )$$
  1. show that $$a = \frac { b ^ { 2 } } { b - 1 }$$ (3)
  2. Write down the full restriction on the value of \(b\), explaining the reason for this restriction.
Edexcel Paper 2 2021 October Q3
3 marks Moderate -0.3
  1. Using the laws of logarithms, solve the equation
$$\log _ { 3 } ( 12 y + 5 ) - \log _ { 3 } ( 1 - 3 y ) = 2$$
AQA C2 2013 January Q4
3 marks Moderate -0.8
4 Given that $$\log _ { a } N - \log _ { a } x = \frac { 3 } { 2 }$$ express \(x\) in terms of \(a\) and \(N\), giving your answer in a form not involving logarithms.
(3 marks)
CAIE P3 2021 March Q1
3 marks Moderate -0.3
Solve the equation \(\ln(x^3 - 3) = 3 \ln x - \ln 3\). Give your answer correct to 3 significant figures. [3]
Edexcel C3 Q25
6 marks Moderate -0.3
  1. Simplify \(\frac{x^2 + 4x + 3}{x^2 + x}\). [2]
  2. Find the value of \(x\) for which \(\log_2 (x^2 + 4x + 3) - \log_2 (x^2 + x) = 4\). [4]
Edexcel C3 Q4
6 marks Moderate -0.3
  1. Simplify \(\frac{x^2 + 4x + 3}{x^2 + x}\). [2]
  2. Find the value of \(x\) for which \(\log_2(x^2 + 4x + 3) - \log_2(x^2 + x) = 4\). [4]