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If $x$ is a real number such that $\log_{3}5=\log_{5}\left ( 2+x \right )$, then which of the following is true?

  1. $0<x<3$
  2. $23<x<30$
  3. $x>30$
  4. $3<x<23$
in Quantitative Aptitude retagged by
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Given that, $\log_{3}{5}=\log_{5}{(x+2)} ;x\in \mathbb{R}$

We know that,  $\log_{3}{5}>\log_{3}{3}$

$\Rightarrow \boxed{\log_{3}{5}>1} \quad [\because \log_{a}a = 1]$

And, $\log_{3}{5}<\log_{3}{9}$

$\Rightarrow \log_{3}{5}<\log_{3}{3^{2}}$

$\Rightarrow \log_{3}{5} < 2 \log_{3}{3} \quad [\because \log_{b}a^{x} = x \log_{b}a]$

$\Rightarrow \boxed{\log_{3}{5}<2}$

So, $ \boxed{1<\log_{3}{5}<2}$

Left-hand side lies between $1$ and $2$. So, the right-hand side also follows the same.

Now,   $ 1<\log_{5}{(x+2)}<2$

$\Rightarrow 5^{1}<x+2<5^{2} \quad \left[\because \log_{a}{x}=b \Rightarrow x=a^{b}\right]$

$\Rightarrow 5<x+2<25 $

$\Rightarrow \boxed{3<x<23}$

Correct Answer $:\text{B}$
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