(a) Write down the set of values of \(x\) for which \(\log_5(12x-4)\) exists.

(b) Solve the equation \(\log_5(12x-4)=\frac{6}{\log_x125}+1\).

Solve the equation \(\log_2(x+1)-4\log_{(x+1)}2=3\).

Solve the equation

\(2\log_p y+10\log_y p-9=0,\)

where \(p\) is a positive constant, giving \(y\) in terms of \(p\).

(a) Express \(2+3\lg x-\lg y\) as a single logarithm to base 10.

(b) (i) Solve \(6x+7-\dfrac3x=0\).

(ii) Hence, given that \(6\log_a3+7-3\log_3a=0\), find the possible values of \(a\).

(i) Show that \(\log_9 4=\log_3 2\).

(ii) Hence solve \(\log_9 4+\log_3 x=3\).