Cho \({\log _b}a = x\) và \({\log _b}c = y\). Hãy biểu diễn \({\log _{{a^2}}}\left( {\sqrt[3]{{{b^5}{c^4}}}} \right)\) theo x và y.

Cho \({\log _b}a = x\) và \({\log _b}c = y\). Hãy biểu diễn \({\log _{{a^2}}}\left( {\sqrt[3]{{{b^5}{c^4}}}} \right)\) theo x và y.
A. \({\log _{{a^2}}}\left( {\sqrt[3]{{{b^5}{c^4}}}} \right) = \frac{{5 + 4y}}{{6x}}\)
B. \({\log _{{a^2}}}\left( {\sqrt[3]{{{b^5}{c^4}}}} \right) = \frac{{20y}}{{3x}}\)
C. \({\log _{{a^2}}}\left( {\sqrt[3]{{{b^5}{c^4}}}} \right) = \frac{{5 + 3y^4}}{{3x^2}}\)
D. \({\log _{{a^2}}}\left( {\sqrt[3]{{{b^5}{c^4}}}} \right) = 20x+\frac{{20y}}{{3}}\)

\({\log _b}a = \frac{{\ln a}}{{\ln b}} = x \Rightarrow \ln a = x.\ln b\left( {a,b > 0} \right)\)
\({\log _b}c = \frac{{{\mathop{\rm lnc}\nolimits} }}{{\ln b}} = y \Rightarrow {\mathop{\rm lnc}\nolimits} = y.\ln b\left( {b,c > 0} \right)\)
\(\begin{array}{l} {\log _{{a^2}}}\left( {\sqrt[3]{{{b^5}{c^4}}}} \right) = \frac{{\ln \left( {\sqrt[3]{{{b^5}{c^4}}}} \right)}}{{\ln \left( {ah2} \right)}} = \frac{{\ln \left( {{b^{\frac{5}{3}}}.{c^{\frac{4}{3}}}} \right)}}{{2.\ln a}} = \frac{{\frac{5}{3}\ln b + \frac{4}{3}\ln c}}{{2.\ln a}}\\ = \frac{{\frac{5}{3}\ln b + \frac{4}{3}y.\ln b}}{{2.x.\ln b}} = \frac{{5 + 4y}}{{6x}} \end{array}\)