Answer
$2\log_5 x+4\log_5y+\log_5 m+\dfrac{1}{3}\log_5 p$
Work Step by Step
$\bf{\text{Solution Outline:}}$
Use the properties of logarithms to rewrite the given expression, $
\log_5 \left(x^2y^4\sqrt[5]{m^3p} \right)
.$
$\bf{\text{Solution Details:}}$
Using the Product Rule of Logarithms, which is given by $\log_b (xy)=\log_bx+\log_by,$ the expression above is equivalent
\begin{array}{l}\require{cancel}
\log_5 x^2+\log_5y^4+\log_5\sqrt[5]{m^3p}
.\end{array}
Using the definition of rational exponents which is given by $a^{\frac{m}{n}}=\sqrt[n]{a^m}=\left(\sqrt[n]{a}\right)^m,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
\log_5 x^2+\log_5y^4+\log_5 (m^3p)^{1/5}
.\end{array}
Using the Power Rule of Logarithms, which is given by $\log_b x^y=y\log_bx,$ the expression above is equivalent
\begin{array}{l}\require{cancel}
2\log_5 x+4\log_5y+\dfrac{1}{3}\log_5 (m^3p)
.\end{array}
Using the Product Rule of Logarithms, which is given by $\log_b (xy)=\log_bx+\log_by,$ the expression above is equivalent
\begin{array}{l}\require{cancel}
2\log_5 x+4\log_5y+\dfrac{1}{3}(\log_5 m^3+\log_5 p)
.\end{array}
Using the Power Rule of Logarithms, which is given by $\log_b x^y=y\log_bx,$ the expression above is equivalent
\begin{array}{l}\require{cancel}
2\log_5 x+4\log_5y+\dfrac{1}{3}(3\log_5 m+\log_5 p)
.\end{array}
Using the Distributive Property which is given by $a(b+c)=ab+ac,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
2\log_5 x+4\log_5y+\dfrac{1}{3}(3\log_5 m)+\dfrac{1}{3}(\log_5 p)
\\\\=
2\log_5 x+4\log_5y+\log_5 m+\dfrac{1}{3}\log_5 p
.\end{array}