The four real numbers $a , b , c$, and $d$ are all greater than 1 .
Suppose that they satisfy the equation $\log _ { c } d = \left( \log _ { a } b \right) ^ { 2 }$.
Use some of the lines given to construct a proof that, in this case, it follows that
$$( * ) \log _ { b } d = \left( \log _ { a } b \right) \left( \log _ { a } c \right)$$
(1) Let $x = \log _ { a } b$ and $y = \log _ { a } c$
(2) $d = \left( c ^ { x } \right) ^ { 2 }$
(3) $d = c ^ { \left( x ^ { 2 } \right) }$
(4) $d = b ^ { x y }$
(5) $d = \left( a ^ { y } \right) ^ { \left( x ^ { 2 } \right) }$
(6) $d = \left( \left( a ^ { y } \right) ^ { x } \right) ^ { 2 }$
(7) $d = \left( a ^ { x } \right) ^ { x y }$
(8) $d = a ^ { \left( y ^ { 2 x } \right) }$
(9) $d = a ^ { \left( x ^ { 2 } y \right) }$