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  4. Let y=√ log 2 3 ⋅ log 2 12 ⋅ log 2 48 ⋅ log 2 192+16- log 2 12 ⋅ log 2 48+10. Find y ∈ N.

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Q. Let $y=\sqrt{\log _2 3 \cdot \log _2 12 \cdot \log _2 48 \cdot \log _2 192+16}-\log _2 12 \cdot \log _2 48+10$. Find $y \in N$.

Continuity and Differentiability

Solution:

$y=\sqrt{\log _2 3 \cdot\left(2+\log _2 3\right) \cdot\left(4+\log _2 3\right) \cdot\left(6+\log _2 3\right)+16}-\left(2+\log _2 3\right) \cdot\left(4+\log _2 3\right)+10$
let $ \log _2 3=x$
$y =\sqrt{x(2+x)(4+x)(6+x)+16}-(2+x) \cdot(4+x)+10 $
$=\sqrt{\left(x^2+6 x\right)\left(x^2+6 x+8\right)+16}-\left(x^2+6 x+8\right)+10$
let $x^2+6 x=t$
$y =\sqrt{ t ( t +8)+16}-( t +8)+10$
$=\sqrt{ t ^2+8 t +16}-( t -2)$
$ = t +4- t +2=6$
hence $y=6$