IIT-JEE 2012 · Paper 1, Mathematics question 57

Self-similar expression. Let $t$ be the argument of the logarithm. Then
$$t=\frac1{3\sqrt2}\sqrt{4-t}\implies18t^2=4-t\implies18t^2+t-4=0.$$
The positive root is $t=\dfrac{-1+17}{36}=\dfrac49$.

Value. $\log_{3/2}\frac49=\log_{3/2}\left(\frac32\right)^{-2}=-2$, so the expression is $6-2=4$.

Answer: 4