Answer
$\log_y \dfrac{\sqrt[6]{p^5}}{p^{2}}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
Use the Laws of Logarithms to write the given expression, $ \dfrac{1}{2}\log_y p^3q^4-\dfrac{2}{3}\log_y p^4q^3 ,$ as a single logarithm.
$\bf{\text{Solution Details:}}$
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} \log_y (p^3q^4)^{1/2}-\log_y (p^4q^3)^{2/3} .\end{array}
Using the Quotient Rule of Logarithms, which is given by $\log_b \dfrac{x}{y}=\log_bx-\log_by,$ the expression above is equivalent
\begin{array}{l}\require{cancel}
\log_y \dfrac{(p^3q^4)^{1/2}}{(p^4q^3)^{2/3}} .\end{array}
Using the extended Power Rule of the laws of exponents which is given by $\left( x^my^n \right)^p=x^{mp}y^{np},$ the expression above is equivalent to \begin{array}{l}\require{cancel}
\log_y \dfrac{p^{3\cdot\frac{1}{2}}q^{4\cdot\frac{1}{2}}}{p^{4\cdot\frac{2}{3}}q^{3\cdot\frac{2}{3}}}
\\\\=
\log_y \dfrac{p^{\frac{3}{2}}q^{2}}{p^{\frac{8}{3}}q^{2}}
\\\\=
\log_y \dfrac{p^{\frac{3}{2}}\cancel{q^{2}}}{p^{\frac{8}{3}}\cancel{q^{2}}}
\\\\=
\log_y \dfrac{p^{\frac{3}{2}}}{p^{\frac{8}{3}}}
.\end{array}
Using the Quotient Rule of the laws of exponents which states that $\dfrac{x^m}{x^n}=x^{m-n},$ the expression above simplifies to
\begin{array}{l}\require{cancel}
\log_y p^{\frac{3}{2}-\frac{8}{3}}
\\\\=
\log_y p^{\frac{9}{6}-\frac{16}{6}}
\\\\=
\log_y p^{-\frac{7}{6}}
.\end{array}
Using the Negative Exponent Rule of the laws of exponents which states that $x^{-m}=\dfrac{1}{x^m}$ or $\dfrac{1}{x^{-m}}=x^m,$ the expression above is equivalent to \begin{array}{l}\require{cancel}
\log_y \dfrac{1}{p^{\frac{7}{6}}}
.\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_y \dfrac{1}{\sqrt[6]{p^7}}
.\end{array}
Rationalizing the denominator results to \begin{array}{l}\require{cancel}
\log_y \dfrac{1}{\sqrt[6]{p^7}}\cdot\dfrac{\sqrt[6]{p^5}}{\sqrt[6]{p^5}}
\\\\=
\log_y \dfrac{\sqrt[6]{p^5}}{\sqrt[6]{p^{12}}}
\\\\=
\log_y \dfrac{\sqrt[6]{p^5}}{\sqrt[6]{(p^{2})^6}}
\\\\=
\log_y \dfrac{\sqrt[6]{p^5}}{p^{2}}
.\end{array}