METHOD 1

${\text{lo}}{{\text{g}}_2}\,y + {\text{lo}}{{\text{g}}_4}\,x + {\text{lo}}{{\text{g}}_4}\,2x = 0$

${\text{lo}}{{\text{g}}_2}\,y + {\text{lo}}{{\text{g}}_4}\,2{x^2} = 0$ M1

${\text{lo}}{{\text{g}}_2}\,y + \frac{1}{2}{\text{lo}}{{\text{g}}_2}\,2{x^2} = 0$ M1

${\text{lo}}{{\text{g}}_2}\,y = - \frac{1}{2}{\text{lo}}{{\text{g}}_2}\,2{x^2}$

${\text{lo}}{{\text{g}}_2}\,y = {\text{lo}}{{\text{g}}_2}\left( {\frac{1}{{\sqrt {2x} }}} \right)$ M1A1

$y = \frac{1}{{\sqrt 2 }}{x^{ - 1}}$ A1

Note: For the final A mark, $y$ must be expressed in the form $p{x^q}$ .

[5 marks]

METHOD 2

${\text{lo}}{{\text{g}}_2}\,y + {\text{lo}}{{\text{g}}_4}\,x + {\text{lo}}{{\text{g}}_4}\,2x = 0$

${\text{lo}}{{\text{g}}_2}\,y + \frac{1}{2}{\text{lo}}{{\text{g}}_2}\,x + \frac{1}{2}{\text{lo}}{{\text{g}}_2}\,2x = 0$ M1

${\text{lo}}{{\text{g}}_2}\,y + {\text{lo}}{{\text{g}}_2}\,{x^{\frac{1}{2}}} + {\text{lo}}{{\text{g}}_2}\,{\left( {2x} \right)^{\frac{1}{2}}} = 0$ M1

${\text{lo}}{{\text{g}}_2}\,\left( {\sqrt 2 xy} \right) = 0$ M1

$\sqrt 2 xy = 1$ A1

$y = \frac{1}{{\sqrt 2 }}{x^{ - 1}}$ A1

Note: For the finalAmark, $y$ must be expressed in the form $p{x^q}$ .

[5 marks]