\(\frac{1}{4}\cdot\frac{2}{6}\cdot\frac{3}{8}\cdot......\cdot\frac{31}{64}=2^x\)
\(\Leftrightarrow\frac{1\cdot2\cdot3\cdot....\cdot31}{4\cdot6\cdot8\cdot....\cdot64}=2^x\)
\(\Leftrightarrow\frac{1\cdot2\cdot3\cdot....\cdot31}{\left(2\cdot2\right)\cdot\left(3\cdot2\right)\cdot\left(4\cdot2\right)\cdot.....\cdot\left(2\cdot32\right)}=2^x\)
\(\Leftrightarrow\frac{1\cdot2\cdot3\cdot...\cdot31}{\left(2\cdot2\cdot2\cdot....\cdot2\right)\left(1\cdot2\cdot3\cdot.....\cdot31\right)\cdot32}=2^x\)
\(\Leftrightarrow\frac{1}{2^{31}.2^5}=2^x\)
\(\Leftrightarrow\frac{1}{2^{36}}=2^x\)
\(\Rightarrow x=-36\)