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 \(b=c\).