Question

The value of $x$ satisfying the equation ${\log }_3(5x-2)-2{\log }_3\sqrt{3x+1}=1-{\log }_{3}4$ is

• A. $2$
• B. $1$
• C. $3$
• D. $4$

Answer 1

Answer: (B)

$1$

${\log }_3(5x-2)-2{\log }_3\sqrt{3x+1}=1-{\log }_{3}4$

L.H.S of the above equation exist when $5x-2>0$ $\Rightarrow x>\frac{2}{5}$

and $3x+1>0$ $\Rightarrow x>-\frac{1}{3}$
Therefore, $x\in (\frac{2}{5},\infty )$
${\log }_3(5x-2)-2{\log }_3\sqrt{3x+1}=1-{\log }_{3}4$
$\Rightarrow {\log }_3(5x-2)-{\log }_3(3x+1)+{\log }_{3}4=1[\because m\log a=log{a}^m]$
$\Rightarrow {\log }_3\frac{4(5x-2)}{(3x+1)}=1[\because \log a+\log b=\log (ab\left)\right]$
$\Rightarrow \frac{4(5x-2)}{(3x+1)}=3^1=3$
$\Rightarrow 20x-8=9x+3$
$\Rightarrow x=1$
Ans: B