## 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 to \begin{array}{l}\require{cancel} \log_b (x^4y^5)^{1/3}-\log_b (x^2y)^{3/4} .\end{array} Using the rule $(ab)^n=a^nb^n$, the expression above is equivalent to: \begin{array}{l}\require{cancel} \log_b {(x^4)^{1/3}(y^5)^{1/3}}-\log_b{(x^2)^{3/4}(y^{3/4})} .\end{array} Using the rule $(a^m)^n=a^{mn}$, the expression above is equivalent to: \begin{array}{l}\require{cancel} \log_b {(x^{4/3}y^{5/3})}-\log_b{(x^{6/4}y^{3/4})} \\=\log_b {(x^{4/3}y^{5/3})}-\log_b{(x^{3/2}y^{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 to \begin{array}{l}\require{cancel} \log_b \dfrac{x^{4/3}y^{5/3}}{ x^{3/2}y^{3/4}} .\end{array} Using the rule $\dfrac{a^m}{a^n} = a^{m-n}$, the expression above is equivalent to: \begin{array}{l}\require{cancel} \log_b{(x^{\frac{4}{3}-\frac{3}{2}} y^{\frac{5}{3}-\frac{3}{4}})} \\=\log_b{(x^{-\frac{1}{6}} y^{\frac{11}{12}})} .\end{array}