Biểu diễn {log_a^2}(sqrt[3](b^5c^4)) theo x={log_b}a và y={log_b}c

${\log _b}a = \frac{{\ln a}}{{\ln b}} = x \Rightarrow \ln a = x.\ln b\left( {a,b > 0} \right)$

${\log _b}c = \frac{{{\mathop{\rm lnc}\nolimits} }}{{\ln b}} = y \Rightarrow {\mathop{\rm lnc}\nolimits} = y.\ln b\left( {b,c > 0} \right)$

$\begin{array}{l} {\log _{{a^2}}}\left( {\sqrt[3]{{{b^5}{c^4}}}} \right) = \frac{{\ln \left( {\sqrt[3]{{{b^5}{c^4}}}} \right)}}{{\ln \left( {ah2} \right)}} = \frac{{\ln \left( {{b^{\frac{5}{3}}}.{c^{\frac{4}{3}}}} \right)}}{{2.\ln a}} = \frac{{\frac{5}{3}\ln b + \frac{4}{3}\ln c}}{{2.\ln a}}\\ = \frac{{\frac{5}{3}\ln b + \frac{4}{3}y.\ln b}}{{2.x.\ln b}} = \frac{{5 + 4y}}{{6x}} \end{array}$