If \( S=\left\{x: \sqrt{\log _{x} \sqrt{3 x}}\right. \), where \( \left.\log _{3} x-1\right\} \)...

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If \( S=\left\{x: \sqrt{\log _{x} \sqrt{3 x}}\right. \), where \( \left.\log _{3} x-1\right\} \), then
(a) \( S \) is a finite set
(b) \( S \in \phi \)
(c) \( S \subset(0, \infty) \)
(d) \( S \) properly contains \( \left(\frac{1}{3}, \infty\right) \)
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