A = {log _{sqrt 2 }}sqrt 6 + {log _4}81 - {log _2}27 + {81^{frac{1}{{{{log }_5}3}}}}

\(\begin{array}{l} A = {\log _{\sqrt 2 }}\sqrt 6 + {\log _4}81 - {\log _2}27 + {81^{\frac{1}{{{{\log }_5}3}}}} = {\log _2}6 + {\log _2}9 - {\log _2}27 + {\left( {{3^{{{\log }_3}5}}} \right)^4}\\ = {\log _2}\frac{{6.9}}{{27}} + {5^4} = 1 + 625 = 626\\ \Rightarrow {\log _2}626 = {\log _2}(2.313) = 1 + {\log _2}313 \end{array}\)