6
\includegraphics[max width=\textwidth, alt={}, center]{436d891d-92ee-4076-8369-db756d413979-2_435_597_1516_776}
The diagram shows the curves \(y = \mathrm { e } ^ { 2 x - 3 }\) and \(y = 2 \ln x\). When \(x = a\) the tangents to the curves are parallel.
- Show that \(a\) satisfies the equation \(a = \frac { 1 } { 2 } ( 3 - \ln a )\).
- Verify by calculation that this equation has a root between 1 and 2 .
- Use the iterative formula \(a _ { n + 1 } = \frac { 1 } { 2 } \left( 3 - \ln a _ { n } \right)\) to calculate \(a\) correct to 2 decimal places, showing the result of each iteration to 4 decimal places.