Express \(g(x)\) in the form \(f(x) + \ln a\) , where \(a \in {{\mathbb{Z}}^ + }\) .

The graph of g is a transformation of the graph of f . Give a full geometric description of this transformation.

attempt to apply rules of logarithms     (M1)

e.g. \(\ln {a^b} = b\ln a\) , \(\ln ab = \ln a + \ln b\)

correct application of \(\ln {a^b} = b\ln a\) (seen anywhere)     A1

e.g. \(3\ln x = \ln {x^3}\)

correct application of \(\ln ab = \ln a + \ln b\) (seen anywhere)     A1

e.g. \(\ln 5{x^3} = \ln 5 + \ln {x^3}\)

so \(\ln 5{x^3} = \ln 5 + 3\ln x\)

\(g(x) = f(x) + \ln 5\) (accept \(g(x) = 3\ln x + \ln 5\) )     A1     N1

[4 marks]

transformation with correct name, direction, and value     A3

e.g. translation by \(\left( {\begin{array}{*{20}{c}}
0\\
{\ln 5}
\end{array}} \right)\) , shift up by \(\ln 5\) , vertical translation of \(\ln 5\)

[3 marks]

This question was very poorly done by the majority of candidates. While candidates seemed to have a vague idea of how to apply the rules of logarithms in part (a), very few did so successfully. The most common error in part (a) was to begin incorrectly with \(\ln 5{x^3} = 3\ln 5x\) . This error was often followed by other errors.

In part (b), very few candidates were able to describe the transformation as a vertical translation (or shift). Many candidates attempted to describe numerous incorrect transformations, and some left part (b) entirely blank.