Question

If $x_0$ satisfies the equation $\log _2\left(x^2-4 x-32\right)=6$ then $\left[\log _{\sqrt{3}} x_0\right]$ equals
[Note: $[ k ]$ denotes greatest integer less than or equal to $k$.]

• A. 2
• B. 3
• C. 4
• D. 5

Answer 1

Answer: (C)

$x^2-4 x-32=64$
$x^2-4 x-96=0 $
$(x-12)(x+8)=0$
$x=12, x=-8 $
$x \neq-8 \Rightarrow x^2-4 x-32=0$
$\log _{\sqrt{3}} x_0=\log _{\sqrt{3}} 12=\log _3 144$
$\log _3 81<\log _3 144<\log _3 243 \Rightarrow 4<\log _3 144<5$
$\therefore \left[\log _3 144\right]=4$