Answer
$\frac{2}{3}\log_5 {x}+\frac{1}{3}\log_5 {y}- \frac{2}{3} $.
Work Step by Step
The given expression is
$=\log_5{\sqrt[3] { \frac{x^2 y}{25} }}$
$=\log_5{\left( \frac{x^2 y}{25} \right )^{\frac{1}{3}} }$
Use power rule.
$=\frac{1}{3} \log_5{\left( \frac{x^2 y}{25} \right ) }$
Use the quotient rule.
$\frac{1}{3} ( \log_5 {x^2y}- \log_5{25} )$
Use the product rule.
$\frac{1}{3} ( \log_5 {x^2}+\log_5 {y}- \log_5{25} )$
$\frac{1}{3} ( \log_5 {x^2}+\log_5 {y}- \log_5{5^2} )$
Use the power rule.
$\frac{1}{3} ( 2\log_5 {x}+\log_5 {y}- 2\log_5{5} )$
Use $\log _aa = 1$.
$\frac{1}{3} ( 2\log_5 {x}+\log_5 {y}- 2 )$
Clear the parentheses.
$\frac{2}{3}\log_5 {x}+\frac{1}{3}\log_5 {y}- \frac{2}{3} $.
