$\ln \left( {\frac{5}{3}} \right)$.

$\ln 45$.

correct approach     (A1)

eg$\,\,\,\,\,$$\ln 5 - \ln 3$

$\ln \left( {\frac{5}{3}} \right) = y - x$    A1     N2

[2 marks]

recognizing factors of 45 (may be seen in log expansion)     (M1)

eg$\,\,\,\,\,$$\ln (9 \times 5),{\text{ }}3 \times 3 \times 5,{\text{ }}\log {3^2} \times \log 5$

correct application of $\log (ab) = \log a + \log b$     (A1)

eg$\,\,\,\,\,$$\ln 9 + \ln 5,{\text{ }}\ln 3 + \ln 3 + \ln 5,{\text{ }}\ln {3^2} + \ln 5$

correct working     (A1)

eg$\,\,\,\,\,$$2\ln 3 + \ln 5,{\text{ }}x + x + y$

$\ln 45 = 2x + y$    A1     N3

[4 marks]

Most candidates were able to earn some or all the marks on this question. Part (a) was answered correctly by nearly all candidates.

Most candidates were able to earn some or all the marks on this question. In part (b), the majority of candidates knew they needed to factor 45, though some did not apply the log rules correctly to earn all the available marks here.