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  4. The value of 6+ log 3 / 2((1/3 √2) √4-(1/3 √2) √4-(1/3 √2) √4-(1/3 √2) ⋅s) is

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Q. The value of $6+\log _{3 / 2}\left(\frac{1}{3 \sqrt{2}} \sqrt{4-\frac{1}{3 \sqrt{2}} \sqrt{4-\frac{1}{3 \sqrt{2}} \sqrt{4-\frac{1}{3 \sqrt{2}} \cdots}}}\right)$ is

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Solution:

Let $\sqrt{4-\frac{1}{3 \sqrt{2}} \sqrt{4-\frac{1}{3 \sqrt{2}} \sqrt{4-\frac{1}{3 \sqrt{2}} \cdots}}}=y$
So, $4-\frac{1}{3 \sqrt{2}} y=y^{2}$
$( y >0)$
$\Rightarrow y^{2}+\frac{1}{3 \sqrt{2}} y-4=0 \Rightarrow y=\frac{8}{3 \sqrt{2}}$
so, the required value is $6+\log _{3 / 2}\left(\frac{1}{3 \sqrt{2}} \times \frac{8}{3 \sqrt{2}}\right)$
$=6+\log _{3 / 2} \frac{4}{9}=6-2=4$