Consider the equation ${\left({\log }_{2}x\right)}^2-4{\log }_{2}x-m^2-2m-13=0$ in the variable $x,$' m '  being a parameter $(m\in R).$ Let the real roots of the equation be $x_1,x_2$ where $x_1<x_2$ then The minimum value of $x_2$ is
 

$$\begin{gathered} {\left({\log }_{2}x\right)}^2-4\left({\log }_{2}x\right)-\left(m^2+2m+13\right)=0 \\ \Rightarrow {\log }_{2}x=\frac{4\pm \sqrt{16+4\left(1\right)\left(m^2+2m+13\right)}}{2} \\ \begin{array}{r}\therefore {\log }_2{x}_1=2-\sqrt{m^2+2m+17}\\ {\log }_2{x}_2=2+\sqrt{m^2+2m+17}\end{array} \\ {\log }_2{x}_2\text{ is minimum when }{x}_2\text{ is minimum, } \\ \begin{array}{r}{\log }_2{x}_2=2+\sqrt{m^2+2m+17}\\ \Rightarrow {\log }_2{x}_2=2+\sqrt{(m+1{)}^2+16}\end{array} \\ \text{ Minimum when }m=-1 \\ \begin{array}{l}{\left({\log }_2{x}_2\right)}_{min.}=2+\sqrt{16}=6\\ \Rightarrow {\left(x_2\right)}_{min.}=2^6=64\end{array} \end{gathered}$$