2025 JPJC P1 Q10

Show that \(y = \ln \left( \frac{e^2}{3x} \right)\) can be written in the form \(y = a + b \ln (cx)\), where \(a\), \(b\) and \(c\) are integers to be found. Hence, state a sequence of transformations which transform the graph of \(y = \ln x\) onto the graph of \(y = \ln \left( \frac{e^2}{3x} \right)\).[4]

The curve \(y = \mathrm{f}(x)\) passes through the point \(P\) with coordinates \((a, b)\), where \(b \neq 0\). The tangent to the curve at \(P\) has gradient \(5\). When \(y = \mathrm{f}(x)\) is transformed onto the curve \(y = \mathrm{g}(x)\), \(P\) corresponds to the point \(R\) on \(y = \mathrm{g}(x)\). For each of the following curves, state the coordinates of \(R\) and find the gradient of the curve at \(R\).