Question

If ${\log }_{3}x+{\log }_{3}y=2+{\log }_{3}2$ and ${\log }_3(x+y)=2$, then the values of $x$ and $y$ are

• A. $x=1,y=8$
• B. $x=8,y=1$
• C. $x=3,y=6$
• D. $x=9,y=3$

Answer 1

The correct option is C $x=3,y=6$

${\log }_{3}x+{\log }_{3}y=2+{\log }_{3}2=2{\log }_{3}3+{\log }_{3}2,[\because {\log }_{3}3=1]$

As,

$[\log a+\log b=\log (ab\left)\right]$ and $[\because x\log a=\log a^x]$

$\Rightarrow {\log }_{3}xy={\log }_3{3}^2+{\log }_{3}2$
$\Rightarrow {\log }_{3}xy={\log }_{3}18$
$\Rightarrow xy=18$ ....(1)

${\log }_3(x+y)=2$
$\Rightarrow x+y=3^2=9$ ....(2)

On solving equation $\left(1\right)$ & $\left(2\right)$, we get
$x=3,y=6$