Given that, $ 2^{x} = 3^{\log_{5}{2}}$
Taking $\log_{2}$ on both sides.
$ \log_{2}{2}^{x} = \log_{2} \left( 3^{\log_{5}{2}} \right) $
$ \Rightarrow x = \log_{5}{2} \log_{2}{3}$ $\quad [\because{ \log_{a}{a}} = 1, \log_{b}{a}^{x} = x \log_{b}{a}$]
$ \Rightarrow x = \frac{\log_{2}{3}}{\log_{2}{5}}$ $\quad \left [ \therefore \log_{a}{b} = \frac{1}{\log_{b}{a}} \right]$
$ \Rightarrow \boxed{x = \log_{5}{3}}$ $\quad \left[ \therefore \frac{\log_{c}{a}}{\log_{c}{b}} = \log_{b}{a} \right]$
Now, we can check all the options.
- $1+\log_{3}{\frac{5}{3}} = 1 + \log_{3}{5} – \log_{3}{3}$
$ \qquad \qquad \quad =1 + \log_{3}{5} – 1 $
$ \qquad \qquad \quad = \log_{3}{5} $
Option $ \text {(B)}$ and $ \text{(D)}$ are not possible.
- $1+\log_{5}{\frac{3}{5}} = 1 + \log_{5}{3} – \log_{5}{5}$
$\qquad \qquad \quad=1 + \log_{5}{3} – 1 $
$\qquad \qquad \quad = \log_{5}{3} $
Correct Answer $: \text{C}$