Question

If $5^{{\left({\log }_{5}7\right)}^{2x}}=7^{{\left({\log }_{7}5\right)}^x}$ then the value of $\frac{1}{x}$ is equal to

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (C)

$5^{{\left({\log }_{5}7\right)}^{2x}}=7^{{\left({\log }_{7}5\right)}^x}\Rightarrow 5^{{\log }_{5}7{\left({\log }_{5}7\right)}^{2x-1}}=7^{{\left({\log }_{7}5\right)}^x}\Rightarrow 7^{{\left({\log }_{5}7\right)}^{2x-1}}=7^{{\left({\log }_{7}5\right)}^x}$
$\Rightarrow {\left({\log }_{5}7\right)}^{2x-1}={\left({\log }_{7}5\right)}^x\Rightarrow {\left({\log }_{5}7\right)}^{2x-1}=\frac{1}{{\left({\log }_{5}7\right)}^x}$
$\therefore {\left({\log }_{5}7\right)}^{3x-1}=1$
$\therefore 3x-1=0\Rightarrow x=\frac{1}{3}$