The four real numbers $a,b,c,$ and $d$ are all greater than 1.

Suppose that they satisfy the equation ${\log }_{c}d=({\log }_{a}b{)}^2$.

Use some of the lines given to construct a proof that, in this case, it follows that

<div align="center"> (*) ${\log }_{b}d=({\log }_{a}b)({\log }_{a}c)$. </div> <br>

(1) Let $x={\log }_{a}b$ and $y={\log }_{a}c$

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(2) $d=(c^x{)}^2$ (3) $d=c^{(x^2)}$ (4) $d=b^{xy}$

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(5) $d=(a^y{)}^{(x^2)}$ (6) $d=((a^y{)}^x{)}^2$ (7) $d=(a^x{)}^{xy}$

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(8) $d=a^{(y^{2x})}$ (9) $d=a^{(x^{2}y)}$