Cho các số thực dương a, b thỏa mãn \({\log _4}a =...

B

- Hướng dẫn giải

\(\begin{array}{l}{\log _4}a = {\log _6}b = {\log _9}\left( {4a - 5b} \right) - 1 = t\\ \Rightarrow \left\{ \begin{array}{l}a = {4^t}\\b = {6^t}\\4a - 5b = {9^{t + 1}}\end{array} \right.\\ \Rightarrow {4.4^t} - {5.6^t} = {9^{t + 1}}\\ \Rightarrow {4.2^{2t}} - {5.2^t}{.3^t} - {9.3^{2t}} = 0\\ \Rightarrow 4.{\left( {\frac{2}{3}} \right)^{2t}} - 5.{\left( {\frac{2}{3}} \right)^t} - 9 = 0\\ \Rightarrow {\left( {\frac{2}{3}} \right)^t} = \frac{9}{4}\\ \Rightarrow t = - 2\\ \Rightarrow P = \frac{b}{a} = \frac{{{6^t}}}{{{4^t}}} = {\left( {\frac{3}{2}} \right)^t} = {\left( {\frac{3}{2}} \right)^{ - 2}} = \frac{4}{9}\end{array}\)