If logk⁡x⋅log5⁡k=logx⁡5,k≠1,k>0 then x is equal to

log k ⁡ x ⋅ log 5 ⁡ k = log x ⁡ 5 or log ⁡ x log ⁡ k log ⁡ k log ⁡ 5 = log x ⁡ 5 or log ⁡ x log ⁡ 5 = log x ⁡ 5 or log 5 ⁡ x = 1 log 5 ⁡ x or log 5 ⁡ x 2 = 1 or log 5 ⁡ x = ± 1 ⇒ x = 5 ± 1 ⇒ x = 1 5 , 5

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If $\log_{k} x \cdot \log_{5} k = \log_{x} 5, k \neq 1, k > 0$ then x is equal to

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1/5

none of these

answer is B, C.

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Detailed Solution

$\begin{array}{ll} \log_{k} x \cdot \log_{5} k = \log_{x} 5 \\ or \frac{\log x}{\log k} \frac{\log k}{\log 5} = \log_{x} 5 \\ or \frac{\log x}{\log 5} = \log_{x} 5 \\ or \log_{5} x = \frac{1}{\log_{5} x} \\ or {(\log_{5} x)}^{2} = 1 \\ or \log_{5} x = \pm 1 \\ \Rightarrow x = 5^{\pm 1} \\ \Rightarrow x = \frac{1}{5}, 5 \end{array}$