Question

The solution set of the equation $4^{{\log }_{9}x}-6.x^{{\log }_{9}2}+2^{{\log }_{3}27}=0$ is _________ .

Answer 1

$x^{{\log }_{x}y}=y$

${\log }_{x}y={\log }_{z}y{\log }_{x}z$

So, $4^{{\log }_{9}x}-6x^{{\log }_{x}2{\log }_{9}x}=-2^3$

$\Rightarrow 4^{{\log }_{9}x}-6\times 2^{{\log }_{9}x}=-8$

Let $2^{{\log }_{9}x}=y$

$\Rightarrow y^2-6y+8=(y-4)(y-2)=0$

So, When $y=4$, ${\log }_{9}x=2\Rightarrow x=81$

When $y=2$, ${\log }_{9}x=1\Rightarrow x=9$

ie, $x\in \{9,81\}$