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