Q. Find the range of ${\log }_3\left\{{\log }_{1/2}\left(x^2+4x+2\right)\right\}$.

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$$\begin{gathered} Let y = {\log }_3\left\{{\log }_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\left(x^2 + 4x + 4\right)\right\} \\ \Rightarrow 3^y = {\log }_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\left(x^2 + 4x + 4\right) \left[\because {\log }_{a}x = b \Rightarrow a^b = x\right] \\ \Rightarrow {\left(\frac{1}{2}\right)}^{3^y} = {\left(x + 2\right)}^2 \left[\because {\log }_{a}x = b \Rightarrow a^b = x\right] \\ \Rightarrow \sqrt{{\left(\frac{1}{2}\right)}^{3^y}} = x + 2 \\ \Rightarrow \sqrt{{\left(\frac{1}{2}\right)}^{3^y}} - 2 = x \\ For the range to be real, {\left(\frac{1}{2}\right)}^{3^y} \ge 0 \\ which is true for all y \in \mathit{R}. \\ So, y can assume the value of any real number, or the range of the given function is \mathit{a}\mathit{l}\mathit{l}\mathbf{ }\mathit{r}\mathit{e}\mathit{a}\mathit{l}\mathbf{ }\mathit{n}\mathit{u}\mathit{m}\mathit{b}\mathit{e}\mathit{r}\mathit{s}\mathbf{.} \end{gathered}$$

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