Answer
$-\displaystyle \frac{2}{3}$
Work Step by Step
From the definition of logarithms,
for $a>0$, $a\neq 1$, and $x > 0$,
$y=\log_{a}x$ means $a^{y}=x$.
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$y=\displaystyle \log_{2}(\sqrt[3]{\frac{1}{4}})\ \ $ means $\ \ 2^{y}=\displaystyle \sqrt[3]{\frac{1}{4}} .$
Since
$\displaystyle \sqrt[3]{\frac{1}{4}}=(\frac{1}{4})^{1/3}=(4^{-1})^{1/3}$
$=(2^{2})^{-1/3}=2^{-2/3},$
$y=-\displaystyle \frac{2}{3}$
