A2. Which expression is equivalent to the expression $\log _{a b}\left(a^{-1} \cdot \sqrt[2]{a^{3} b^{-1}}: \sqrt[3]{b^{-1} a^{2}}\right)$ ?
(A) $a b$
(B) $\frac{a}{b}$
(C) 1
(D) $-\frac{1}{6}$
(E) $-\frac{1}{2}$
A2. The expression in parentheses is transformed to a common root exponent $a^{-1} \cdot \sqrt[2]{a^{3} b^{-1}}: \sqrt[3]{b^{-1} a^{2}}=$ $\sqrt[6]{a^{-6}} \cdot \sqrt[6]{a^{9} b^{-3}}: \sqrt[6]{b^{-2} a^{4}}$. We simplify the expression and put it under a common root: $\sqrt[6]{a^{-1} b^{-1}}=\sqrt[6]{\frac{1}{a b}}=$ $\left(\frac{1}{a b}\right)^{\frac{1}{6}}=a b^{-\frac{1}{6}}$. The logarithm is rearranged to $\log _{a b}\left(a^{-1} \cdot \sqrt[2]{a^{3} b^{-1}}: \sqrt[3]{b^{-1} a^{2}}\right)=\log _{a b}(a b)^{-\frac{1}{6}}=-\frac{1}{6}$.