Question

If $2^{x+y}=6^y$, then the value of $\frac{x}{y}$ is

• A. $2{\log }_{2}3$
• B. ${\log }_{3}2$
• C. $2{\log }_{3}2$
• D. ${\log }_{2}3$

Answer 1

The correct option is D ${\log }_{2}3$

$2^{x+y}=6^y\Rightarrow 2^x\cdot 2^y=2^y\cdot 3^y\Rightarrow 2^y(2^x-3^y)=0\Rightarrow 2^x=3^y(\because 2^y\ne 0)$
Taking ${\log }_2$ on both sides, we get
$\Rightarrow x{\log }_{2}2=y{\log }_{2}3\Rightarrow \frac{x}{y}={\log }_{2}3$