Logarithms

The value of $6+\log_{3/2}\left(\dfrac1{3\sqrt2}\sqrt{4-\dfrac1{3\sqrt2}\sqrt{4-\dfrac1{3\sqrt2}\sqrt{4-\dfrac1{3\sqrt2}\cdots}}}\right)$ is

Let $(x_0,y_0)$ be the solution of the following equations
$$(2x)^{\ln2}=(3y)^{\ln3},\qquad3^{\ln x}=2^{\ln y}.$$
Then $x_0$ is

Match the Statements / Expressions in Column I with the Statements / Expressions in Column II and indicate your answer by darkening the appropriate bubbles in the 4 × 4 matrix given in the ORS.