June 2025 Paper 2 Q3

Notes

Note: Condone absence of reference to the value of \(\dfrac{x}{y}\) being undefined when \(x = y = 0\)

M1: For the key step in attempting to take logarithms with the same base of both sides and apply the power law to both sides, e.g., \(x = y\log_3 7\) or \(y = x\log_7 3\) or \(x\ln 3 = y\ln 7\)
Condone e.g. \(x = \log_3 7y\) but not \(x = \log_3(7y)\)
Alternatively, takes a root of both sides (either \(x\) or \(y\)), takes logs with the same base of both sides and applies the power law e.g., \(3^x = 7^y \rightarrow 3^{\frac{x}{y}} = 7 \rightarrow \dfrac{x}{y}\log 3 = \log 7\)
May be implied by a correct answer.

A1: \(\left(\dfrac{x}{y} =\right) \dfrac{\log 7}{\log 3}\) or equivalent, e.g, \(\dfrac{\ln 7}{\ln 3}\) or \(\log_3 7\) or \(\dfrac{1}{\log_7 3}\) Condone e.g. \(\dfrac{\ln|7|}{\ln|3|}\)
Correct answer only scores both marks provided there is no incorrect log work.
Any base \(k\) may be used for e.g. \(\dfrac{\log_k 7}{\log_k 3}\) provided it is the same base in both logs, although watch out for e.g. \(\left(\dfrac{\log_3 7}{\log_7 3}\right)^{\frac{1}{2}}\) or \(\dfrac{\log_3 k}{\log_7 k}\) for constant \(k \gt 0\) which are exceptions and correct.
There is no need to see \(\dfrac{x}{y} =\) but it should be clear what their answer is.
Do not ISW if they incorrectly apply log laws e.g. \(\dfrac{\log 7}{\log 3} = \log\dfrac{7}{3}\) or \(= \log 7 - \log 3\) or \(= \log(7 - 3)\) all of which score M1A0.
You may ISW after a correct answer if they go on to provide a decimal approximation.
A decimal approximation with no correct log work seen (1.7712…) scores no marks.