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  4. If 5( log 5 7)2 x=7( log 7 5)x then the value of (1/x) is equal to

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Q. If $5^{\left(\log _5 7\right)^{2 x}}=7^{\left(\log _7 5\right)^x}$ then the value of $\frac{1}{x}$ is equal to

Continuity and Differentiability

Solution:

$5^{\left(\log _5 7\right)^{2 x}}=7^{\left(\log _7 5\right)^x} \Rightarrow 5^{\log _5 7\left(\log _5 7\right)^{2 x-1}}=7^{\left(\log _7 5\right)^x} \Rightarrow 7^{\left(\log _5 7\right)^{2 x-1}}=7^{\left(\log _7 5\right)^x} $
$\Rightarrow \left(\log _5 7\right)^{2 x-1}=\left(\log _7 5\right)^x \Rightarrow\left(\log _5 7\right)^{2 x-1}=\frac{1}{\left(\log _5 7\right)^x} $
$\therefore \left(\log _5 7\right)^{3 x-1}=1 $
$\therefore 3 x-1=0 \Rightarrow x=\frac{1}{3}$