2 Use logarithms to solve the equation \(6 ^ { 2 x - 1 } = 5 e ^ { 3 x + 2 }\). Give your answer correct to 4 significant figures. [4]
\includegraphics[max width=\textwidth, alt={}, center]{6ee58f43-831d-402c-9f9a-2b247b2f7ffc-04_778_486_276_769} The diagram shows the curve with equation \(\mathrm { y } = 8 \mathrm { e } ^ { - \mathrm { x } } - \mathrm { e } ^ { 2 \mathrm { x } }\). The curve crosses the \(y\)-axis at the point \(A\) and the \(x\)-axis at the point \(B\). The shaded region is bounded by the curve and the two axes.

1. Find the gradient of the curve at \(A\).
2. Show that the \(x\)-coordinate of \(B\) is \(\ln 2\) and hence find the area of the shaded region.