Let \(f(x) = \ln x\) . The graph of f is transformed into the graph of the function g by a translation of \(\left( {\begin{array}{*{20}{c}}
  3 \\
  { - 2}
\end{array}} \right)\), followed by a reflection in the x-axis. Find an expression for \(g(x)\), giving your answer as a single logarithm.

\(h(x) = f(x - 3) - 2 = \ln (x - 3) - 2\)     (M1)(A1)

\(g(x) =  -h(x) = 2 - \ln (x - 3)\)     M1

 Note: Award M1 only if it is clear the effect of the reflection in the x-axis:

the expression is correct OR

there is a change of signs of the previous expression OR

there’s a graph or an explanation making it explicit

\( = \ln {{\text{e}}^2} - \ln (x - 3)\)     M1

\( = \ln \left( {\frac{{{{\text{e}}^2}}}{{x - 3}}} \right)\)     A1

[5 marks]

This question was well attempted but many candidates could have scored better had they written down all the steps to obtain the final expression. In some cases, as the final expression was incorrect and the middle steps were missing, candidates scored just 1 mark. That could be a consequence of a small mistake, but the lack of working prevented them from scoring at least all method marks. Some candidates performed the transformations well but were not able to use logarithms properties to transform the answer and give it as a single logarithm.