Question

If the least integral value satisfying the equation $\log _3 \sqrt{x^2-4 x+4}=2^{\log _2\left(\log _3(|x|-2)\right)}$ is $\alpha$, then find the number of zeroes after decimal and before first significant digit in the number of $(\alpha)^{-4 \alpha}$.

Answer 1

$ \log _3| x -2|=2^{\log _2\left(\log _3(| x |-2)\right)}$
$\Rightarrow \log _3| x -2|=\log _3(| x |-2)$
$\Rightarrow| x -2|=| x |-2$
$\Rightarrow| x |-| x -2|=2$
$\text { Case-I }: x <0$
$- x + x -2=0 $
$-2=2 \text { (no solution) }$
Case-II: $0 \leq x < 2$
$x+(x-2)=2 $
$\Rightarrow 2 x=4 \Rightarrow x=2 \text { (no solution) }$
$\text { Case-III }: x \geq 2$
$x-(x-2)=2 $
$\Rightarrow 2=2$
$\therefore x \geq 2$
But $\log _3(|x|-2)>0 \Rightarrow|x|-2>1 \Rightarrow|x|>3$
$\therefore x \in(3, \infty)$ is the solution set of the equation
Least integral value of $x$ is 4 .
Now,$(\alpha)^{-4 \alpha}=(4)^{-16}=2^{-32} $
$N=2^{-32} $
$\log _{10} N =-32 \log _{10} 2=-32 \times 0.301=-9.632$
$\therefore$ Number of zeros after decimal and before first significant digit in $2^{-32}$ is 9