Question

If ${\log }_3(3x-8)=2-x,$ then the value of $x$ is

• A. $-1$
• B. $2$
• C. $9$
• D. $3$

Answer 1

The correct option is B $2$

${\log }_3(3^x-8)=2-x$
For the $\log$ to be defined, we get
$3^x-8>0$
$\Rightarrow 3^x>8$
Taking ${\log }_3$ on both sides, we get
$\Rightarrow x>{\log }_{3}8$

Now,
${\log }_3(3^x-8)=2-x\Rightarrow 3^x-8=3^{2-x}\Rightarrow 3^x-8=\frac{3^2}{3^x}$

Assuming $3^x=t$
$\Rightarrow t-8=\frac{9}{t}\Rightarrow t^2-8t-9=0\Rightarrow (t-9)(t+1)=0\Rightarrow t=-1,9\Rightarrow 3^x=9(\because 3^x>0)\therefore x=2$