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.