By sketching a suitable pair of graphs, show that the equation
$$\ln x = 4 - \frac { 1 } { 2 } x$$
has exactly one real root, \(\alpha\).
Verify by calculation that \(4.5 < \alpha < 5.0\).
Use the iterative formula \(x _ { n + 1 } = 8 - 2 \ln x _ { n }\) to find \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.