Write \(\ln ({x^2} - 1) - 2\ln (x + 1) + \ln ({x^2} + x)\) as a single logarithm, in its simplest form.

\(\ln ({x^2} - 1) - \ln {(x + 1)^2} + \ln x(x + 1)\)     (A1)

\( = \ln \frac{{x({x^2} - 1)(x + 1)}}{{{{(x + 1)}^2}}}\)     (M1)A1

\( = \ln \frac{{x(x + 1)(x - 1)(x + 1)}}{{{{(x + 1)}^2}}}\)     (A1)

\( = \ln x(x - 1)\,\,\,\,\,\left( { = \ln ({x^2} - x)} \right)\)     A1

[5 marks]

There were fewer correct solutions to this question than might be expected. A significant number of students managed to combine the terms to form one logarithm, but rather than factorising, then expanded the brackets, which left them unable to gain an answer in its simplest form.