Cho \(a{\log _2}3 + b{\log _6}2 + c{\log _6}3 = 5\...

D

- Hướng dẫn giải

$a{\log _2}3 + b{\log _6}2 + c{\log _6}3 = 5 \Leftrightarrow {\log _6}{2^b} + {\log _6}{3^c} = {\log _2}{2^5} - {\log _2}{3^a}$ $\Leftrightarrow {\log _6}{2^b}{3^c} = {\log _2}\frac{{{2^5}}}{{{3^a}}}$.

Đặt $\left\{ \begin{array}{l} t = {\log _6}{2^b}{3^c}\\ t = {\log _2}\frac{{{2^5}}}{{{3^a}}} \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} {2^b}{3^c} = {6^t}\\ \frac{{{2^5}}}{{{3^a}}} = {2^t} \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} {2^b}{3^c} = {6^t}\\ {2^5} = {3^a}{2^t} \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} a = 0\\ t = 5\\ b = c = 5 \end{array} \right.$ (vì $a, b,c$ là các số tự nhiên).

Vậy $b=c$.