4 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } + 3 x ^ { 2 } + 4 a x - 5 ,$$ where \(a\) is a constant. It is given that ( \(2 x - 1\) ) is a factor of \(\mathrm { p } ( x )\).

1. Use the factor theorem to find the value of \(a\).
2. Factorise \(\mathrm { p } ( x )\) and hence show that the equation \(\mathrm { p } ( x ) = 0\) has only one real root.
3. Use logarithms to solve the equation \(\mathrm { p } \left( 6 ^ { y } \right) = 0\) correct to 3 significant figures.