Question

The absolute difference between the roots of the equation $4^x-3\left(2^{x+3}\right)+108=0$ is

• A. $12$
• B. $1$
• C. ${\log }_{3}2$
• D. ${\log }_{2}3$

Answer 1

The correct option is D ${\log }_{2}3$

$4^x-3\left(2^{x+3}\right)+108=0$
Assuming $2^x=y$
$y^2-3\left(2^{3}y\right)+108=0\Rightarrow y^2-24y+108=0$
$\Rightarrow (y-6)(y-18)=0\Rightarrow y=18,6$
$\Rightarrow 2^x=18,6$
Taking ${\log }_2$ both sides, we get
$\Rightarrow x={\log }_{2}18,{\log }_{2}6$
$\therefore$ Absolute difference of the roots is
${\log }_{2}18-{\log }_{2}6={\log }_{2}3$