Exponential to linear form proof

Given an equation like a^y = b^x, prove using logarithms that it can be written as y = kx and find the constant k.

7 questions · Moderate -0.9

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CAIE P2 2020 June Q1

3 marks Moderate -0.8

1 Given that \(2 ^ { y } = 9 ^ { 3 x }\), use logarithms to show that \(y = k x\) and find the value of \(k\) correct to 3 significant figures.

CAIE P2 2024 November Q1

3 marks Moderate -0.8

1 Use logarithms to show that the equation \(5 ^ { 8 y } = 6 ^ { 7 x }\) can be expressed in the form \(y = k x\). Give the value of the constant \(k\) correct to 3 significant figures.

CAIE P2 2004 June Q1

3 marks Easy -1.2

1 Given that \(2 ^ { x } = 5 ^ { y }\), use logarithms to find the value of \(\frac { x } { y }\) correct to 3 significant figures.

CAIE P2 2009 June Q1

3 marks Moderate -0.8

1 Given that \(( 1.25 ) ^ { x } = ( 2.5 ) ^ { y }\), use logarithms to find the value of \(\frac { x } { y }\) correct to 3 significant figures.

CAIE P2 2010 June Q1

3 marks Moderate -0.8

1 Given that \(13 ^ { x } = ( 2.8 ) ^ { y }\), use logarithms to show that \(y = k x\) and find the value of \(k\) correct to 3 significant figures.

CAIE P2 2016 June Q1

3 marks Moderate -0.8

1 Given that \(5 ^ { 3 x } = 7 ^ { 4 y }\), use logarithms to find the value of \(\frac { x } { y }\) correct to 4 significant figures.

CAIE P2 2017 June Q1

3 marks Moderate -0.8

1 Given that \(5 ^ { x } = 3 ^ { 4 y }\), use logarithms to show that \(y = m x\) and find the value of the constant \(m\) correct to 3 significant figures.