Question

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

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is A $1$

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

As ${\log }_{c}a-{\log }_{c}b={\log }_c\left(\frac{a}{b}\right)$

& $n{\log }_{c}m={\log }_{c}mn$

& ${\log }_{c}c=1$

$\Rightarrow {\log }_3\left[\frac{4}{(\sqrt{3x+1}{)}^2}\right]={\log }_{3}3-{\log }_3(5x-2)$

$\Rightarrow {\log }_3\left(\frac{4}{3x+1}\right)={\log }_3\left(\frac{3}{5x-2}\right)$

Taking & antilog

$\Rightarrow \frac{4}{3x+1}=\frac{3}{5x-2}$

$\Rightarrow 20x-8=9x+3$

$\Rightarrow x=1$