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$The number of roots of the equation 3 x^{\log_{5} 2} + 2^{\log_{5} x} = 64 is / are$

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Detailed Solution

$3 x^{{\log_{5}}^{2}} + 2^{{\log_{5}}^{x}} = 64$

$\Rightarrow 3 x^{\log_{5} {}^{2}} + x^{\log_{5} 2} = 64$ $\{∵ a^{\log_{c} b} = b^{\log_{c} a}\}$

$\Rightarrow 4 x^{\log_{5} 2} = 64 \Rightarrow x^{\log_{5} 2} = 16$

$\Rightarrow 2^{\log_{5} x} = 2^{4} \Rightarrow \log_{5} x = 4$

$\Rightarrow x = 5^{4} = 625$

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