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$a$/ Nếu $a > 1\,\,$ta có: $\,\,0 < {\log _a}2 < {\log _a}3$
$\begin{array}{l}
\Rightarrow \frac{1}{{{{\log }_2}a}} < \frac{1}{{{{\log }_3}a}}\\
\Rightarrow {\log _2}a > {\log _3}a
\end{array}$
Tương tự, nếu $0 < a < 1:\,\,0 > {\log _a}2 > {\log _a}3$
$\begin{array}{l}
\Rightarrow \frac{1}{{{{\log }_2}a}} > \frac{1}{{{{\log }_3}a}}\\
\Rightarrow {\log _2}a < {\log _3}a
\end{array}$ Nếu $a = 1\,$ta có ${\log _2}a = {\log _3}a = 0$
$\begin{array}{l}
b/\,Ta\,\,co'\,\,3 = \sqrt 9 > \sqrt 8 = {2^{\frac{3}{2}}} \Rightarrow {\log _2}3 > \frac{3}{2}\\
\Rightarrow {3^{{{\log }_2}3}} > {3^{3/2}} = \sqrt {27} > 5 \Rightarrow {\log _2}3 > {\log _3}5
\end{array}$
$c$/ Đổi về cơ số $10$ ta có:
$\begin{array}{l}
X = {\log _{135}}675 = \frac{{\log 675}}{{\log 135}} = \frac{{\log 9.75}}{{\log 3.45}} = \frac{{2\log 3 + \log 75}}{{\log 3 + \log 45}}\\
Y = {\log _{45}}75 = \frac{{\log 75}}{{\log 45}}\\
X - Y = \frac{{2\log 3 + \log 75}}{{\log 3 + \log 45}} - \frac{{\log 75}}{{\log 45}} = \frac{{\log 3\left( {2\log 45 - \log 75} \right)}}{{\left( {\log 3 + \log 45} \right)\log 45}}\\
= \frac{{\log 3\left( {{{\log }^2}45 - \log 75} \right)}}{{\left( {\log 3 + \log 45} \right)\log 45}} > 0 \Rightarrow x > y\\
suy\,\,ra\,\,:\,\,{\log _{135}}675 > {\log _{45}}75
\end{array}$
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