Nếu \({{\log }_{2}}\left( {{\log }_{8}}x \right)={{\log }_{8}}\left( {{\log }_{2}}x \right)\) thì \({{\left( {{\log }_{2}}x \right)}^{2}}\) bằng

* Hướng dẫn giải

ĐK: \(\left\{ \begin{array}{l}
x > 0\\
{\log _2}x > 0\\
{\log _8}x > 0
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
x > 0\\
x > 1
\end{array} \right..\)

\(Pt\Leftrightarrow {{\log }_{2}}\left( {{\log }_{8}}x \right)=\frac{1}{3}{{\log }_{2}}\left( {{\log }_{2}}x \right)\Leftrightarrow \frac{1}{3}{{\log }_{2}}x={{\left( {{\log }_{2}}x \right)}^{\frac{1}{3}}}.\)

\(\begin{array}{l}
\Leftrightarrow \frac{1}{{27}}\log _2^3x = {\log _2}x \Leftrightarrow \log _2^3x - 27{\log _2}x = 0\\
\Leftrightarrow \left[ \begin{array}{l}
{\log _2}x = 0\;\;\left( {ktm} \right)\\
\log _2^2x = 27
\end{array} \right. \Rightarrow \log _2^2x = 27.
\end{array}\)

Chọn D