8. The curve with equation \(y = \ln 3 x\) crosses the \(x\)-axis at the point \(P ( p , 0 )\).
- Sketch the graph of \(y = \ln 3 x\), showing the exact value of \(p\).
The normal to the curve at the point \(Q\), with \(x\)-coordinate \(q\), passes through the origin.
- Show that \(x = q\) is a solution of the equation \(x ^ { 2 } + \ln 3 x = 0\).
- Show that the equation in part (b) can be rearranged in the form \(x = \frac { 1 } { 3 } \mathrm { e } ^ { - x ^ { 2 } }\).
- Use the iteration formula \(x _ { n + 1 } = \frac { 1 } { 3 } \mathrm { e } ^ { - x _ { n } ^ { 2 } }\), with \(x _ { 0 } = \frac { 1 } { 3 }\), to find \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\). Hence write down, to 3 decimal places, an approximation for \(q\).