Question

Solve ${\log }_4(2\times 4^{x-2}-1)+4=2x$

Answer 1

${\log }_4(2\times 4^{x-2}-1)+4=2x$
${\log }_4(2\times 4^{x-2}-1)=2x-4$
$\Rightarrow 4^{2x-4}=2\times 4^{x-2}-1$
$4^{2(x-2)}-2\times 4^{x-2}+1=0$
Put $4^{x-2}=t$
$\Rightarrow t^2-2t+1=0$
$\Rightarrow (t-1{)}^2=0$
$\Rightarrow t=1$
$\Rightarrow 4^{x-2}=1$
$\Rightarrow x-2=0$
$\Rightarrow x=2$
From the given equation , it follows that
$2\times 4^{x-2}-1>0$
$\Rightarrow 4^{x-2}>\frac{1}{2}$
$\Rightarrow 2^{2x-4}>2^{-1}$
$\Rightarrow x>\frac{3}{2}$
So, $x=2$ is the solution of the given eqn