## College Algebra (11th Edition)

$\log_b \dfrac{\sqrt[3]{x^{4}y^{5}}}{ \sqrt[4]{x^{6}y^{3}}}$
$\bf{\text{Solution Outline:}}$ Use the Laws of Logarithms to write the given expression, $\dfrac{1}{3}\log_b x^4y^5-\dfrac{3}{4}\log_b x^2y ,$ 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_b (x^4y^5)^{1/3}-\log_b (x^2y)^{3/4} .\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_b \dfrac{(x^4y^5)^{1/3}}{ (x^2y)^{3/4}} .\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_b \dfrac{\sqrt[3]{(x^4y^5)^1}}{ \sqrt[4]{(x^2y)^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_b \dfrac{\sqrt[3]{x^{4(1)}y^{5(1)}}}{ \sqrt[4]{x^{2(3)}y^{1(3)}}} \\\\= \log_b \dfrac{\sqrt[3]{x^{4}y^{5}}}{ \sqrt[4]{x^{6}y^{3}}} .\end{array}