Solve using substitution or auxiliary variable

1. (i) Given that

$$\log _ { a } x + \log _ { a } 3 = \log _ { a } 27 - 1 , \text { where } a \text { is a positive constant }$$ find, in its simplest form, an expression for \(x\) in terms of \(a\).
(ii) Solve the equation $$\left( \log _ { 5 } y \right) ^ { 2 } - 7 \left( \log _ { 5 } y \right) + 12 = 0$$ showing each step of your working.

5. (a) Given that $$y = \log _ { 3 } x$$ find expressions in terms of \(y\) for

1. \(\log _ { 3 } \left( \frac { x } { 9 } \right)\)
2. \(\log _ { 3 } \sqrt { x }\) Write each answer in its simplest form.
   (b) Hence or otherwise solve $$2 \log _ { 3 } \left( \frac { x } { 9 } \right) - \log _ { 3 } \sqrt { x } = 2$$

5. (a) Given that \(t = \log _ { 3 } x\), find expressions in terms of \(t\) for

1. \(\log _ { 3 } x ^ { 2 }\),
2. \(\log _ { 9 } x\).
   (b) Hence, or otherwise, find to 3 significant figures the value of \(x\) such that $$\log _ { 3 } x ^ { 2 } - \log _ { 9 } x = 4 .$$

3. (a) Given that \(y = \log _ { 2 } x\), find expressions in terms of \(y\) for

1. \(\quad \log _ { 2 } \left( \frac { x } { 2 } \right)\),
2. \(\log _ { 2 } ( \sqrt { x } )\).
   (b) Hence, or otherwise, solve the equation $$2 \log _ { 2 } \left( \frac { x } { 2 } \right) + \log _ { 2 } ( \sqrt { x } ) = 8$$