Sketch the curve with equation \(y = 4 ^ { x }\), indicating the coordinates of any point where the curve intersects the coordinate axes.
(2 marks)
Describe the geometrical transformation that maps the graph of \(y = 4 ^ { x }\) onto the graph of \(y = 4 ^ { x } - 5\).
Use the substitution \(Y = 2 ^ { x }\) to show that the equation \(4 ^ { x } - 2 ^ { x + 2 } - 5 = 0\) can be written as \(Y ^ { 2 } - 4 Y - 5 = 0\).
Hence show that the equation \(4 ^ { x } - 2 ^ { x + 2 } - 5 = 0\) has only one real solution. Use logarithms to find this solution, giving your answer to three decimal places.
(4 marks)