Answer
$\frac{1}{5}\log_2 {x}+\frac{4}{5}\log_2 {y}- \frac{4}{5} $.
Work Step by Step
The given expression is
$=\log_2{\sqrt[5] { \frac{x y^4}{16} }}$
$=\log_2{\left( \frac{x y^4}{16} \right )^{\frac{1}{5}} }$
Use power rule.
$=\frac{1}{5} \log_2{\left( \frac{x y^4}{16} \right ) }$
Use the quotient rule.
$=\frac{1}{5} ( \log_2 {xy^4}- \log_2{16} )$
Use the product rule.
$=\frac{1}{5} ( \log_2 {x}+\log_2 {y^4}- \log_2{16} )$
$=\frac{1}{5} ( \log_2 {x}+\log_2 {y^4}- \log_2{2^4} )$
Use the power rule.
$=\frac{1}{5} ( \log_2 {x}+4\log_2 {y}- 4\log_2{2} )$
Use $\log _aa = 1$.
$=\frac{1}{5} ( \log_2 {x}+4\log_2 {y}- 4 )$
Clear the parentheses.
$=\frac{1}{5}\log_2 {x}+\frac{4}{5}\log_2 {y}- \frac{4}{5} $.
