Note that if we set \(log_{\sqrt{2}}4 = x_{1}, \hspace{1mm}solving\hspace{1mm}gives\hspace{1mm}x_{1}=4\\if\hspace{1mm}log_{\frac{1}{2}}16 = x_{2}, \Rightarrow x_{2} = -4\\Also \hspace{1mm}for\hspace{1mm}log_{4}32 = x_{3} = 2.5\\Combining\hspace{1mm}results,\hspace{1mm}x_{1}+x_{2}+x_{3} = 4+(-4)-2.5=-2.5\)
Note that the solution has been separated to simplify solving difficulty due to different bases.