Question

Find the value of $x$, if ${\log }_2(4\times 3^x-6)-{\log }_2(9^x-6)=1$

Answer 1

${\log }_2(4\times 3^x-6)-{\log }_2(9^x-6)=1$.
$\Rightarrow$ ${\log }_2\frac{4\times 3^x-6}{9^x-6}=1[\because \log a-\log b=\frac{a}{b}]$
$\Rightarrow$ $\frac{4\times 3^x-6}{9^x-6}=2$
Substituting, $3^x=y$, we get
$\Rightarrow$ $4y-6=2y^2-12$
$\Rightarrow$ $y^2-2y-3=0$
$\Rightarrow$ $y=-1,3$
$\Rightarrow$ $3^x=3$
$\Rightarrow$ $x=1$
Ans: 1