Cho các số thực dương a, b thỏa mãn \({\log _{16}}...

C

- Hướng dẫn giải

Đặt \({\log _{16}}a = {\log _{20}}b = {\log _{25}}\frac{{2a - b}}{3} = t \Rightarrow \left\{ \begin{array}{l}
a = {16^t}\\
b = {20^t}\\
2a - b = {3.25^t}
\end{array} \right. \Rightarrow \left\{ \begin{array}{l}
{2.16^t} - {20^t} = {3.25^t}(1)\\
\frac{a}{b} = {\left( {\frac{4}{5}} \right)^t}
\end{array} \right.\) 

\(\left( 1 \right) \Leftrightarrow 2.{\left( {\frac{{16}}{{25}}} \right)^t} - {\left( {\frac{4}{5}} \right)^t} - 3 = 0 \Leftrightarrow 2.{\left( {\frac{4}{5}} \right)^{2t}} - {\left( {\frac{4}{5}} \right)^t} - 3 = 0 \Leftrightarrow \left[ \begin{array}{l}
{\left( {\frac{4}{5}} \right)^t} =  - 1 < 0\\
{\left( {\frac{4}{5}} \right)^t} = \frac{3}{2}
\end{array} \right. \Leftrightarrow {\left( {\frac{4}{5}} \right)^t} = \frac{3}{2}\) 

\( \Rightarrow T = \frac{a}{b} = \frac{3}{2} \Rightarrow 1 < T < 2.\)