a. \(2^{2\log_25+\log_{\frac{1}{2}}9}\) và \(\frac{\sqrt{626}}{6}\)
Ta có : \(2^{2\log_25+\log_{\frac{1}{2}}9}=2^{\log_225-\log_29}=2^{\log_2\frac{25}{9}}=\frac{25}{9}=\frac{\sqrt{625}}{9}< \frac{\sqrt{626}}{6}\)
\(\Rightarrow2^{2\log_25+\log_{\frac{1}{2}}9}< \frac{\sqrt{626}}{6}\)
b. \(3^{\log_61,1}\) và \(7^{\log_60,99}\)
Ta có : \(\begin{cases}\log_61,1>0\Rightarrow3^{\log_61,1}>3^0=1\\\log_60,99< 0\Rightarrow7^{\log_60,99}< 7^0=1\end{cases}\)
\(\Rightarrow3^{\log_61,1}>7^{\log_60,99}\)
c. \(\log_{\frac{1}{3}}\frac{1}{80}\) và \(\log_{\frac{1}{2}}\frac{1}{15+\sqrt{2}}\)
Ta có : \(\begin{cases}\log_{\frac{1}{2}}\frac{1}{80}=\log_{3^{-1}}80^{-1}=\log_380< \log_381=4\\\log_{\frac{1}{2}}\frac{1}{15+\sqrt{2}}=\log_{2^{-1}}\left(15+\sqrt{2}\right)^{-1}=\log_2\left(15+\sqrt{2}\right)>\log_216=4\end{cases}\)
\(\Rightarrow\log_{\frac{1}{3}}\frac{1}{80}< \log_{\frac{1}{2}}\frac{1}{15+\sqrt{2}}\)
