Find x if log₂log_1/2log₃x > 0.............. plzz explain the steps as well..

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Please find below the solution to the asked query:

$$\begin{gathered} {\log }_2{\log }_{\frac{1}{2}}{\log }_3\mathrm{x}>0 \\ \Rightarrow {\log }_{\frac{1}{2}}{\log }_3\mathrm{x}>2^0 \\ \left(\mathrm{As} \log \mathrm{with} \mathrm{base} 2 \mathrm{is} \mathrm{an} \mathrm{increasing} \mathrm{function}, \mathrm{hence} \mathrm{direction} \mathrm{of} \mathrm{inequality} \mathrm{will} \mathrm{remain} \mathrm{same}.\right) \\ \Rightarrow {\log }_{\frac{1}{2}}{\log }_3\mathrm{x}>1 \\ \Rightarrow {\log }_3\mathrm{x}<{\left(\frac{1}{2}\right)}^1 \\ \left(\mathrm{As} \log \mathrm{with} \mathrm{base} \frac{1}{2} \mathrm{is} \mathrm{a} \mathrm{decreasing} \mathrm{function}, \mathrm{hence} \mathrm{direction} \mathrm{of} \mathrm{inequality} \mathrm{will} \mathrm{reverse}.\right) \\ \Rightarrow {\log }_3\mathrm{x}<\frac{1}{2} \\ \Rightarrow \mathrm{x}<3^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} \\ \left(\mathrm{As} \log \mathrm{with} \mathrm{base} 3 \mathrm{is} \mathrm{an} \mathrm{increasing} \mathrm{function}, \mathrm{hence} \mathrm{direction} \mathrm{of} \mathrm{inequality} \mathrm{will} \mathrm{remain} \mathrm{same}.\right) \\ \Rightarrow \mathrm{x}<\sqrt{3} \left(\mathrm{Answer}\right) \end{gathered}$$

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