Question

Sắp xếp theo thứ tự giảm dần :

\(2\log_45;\log_3\frac{\pi}{4};\log_{\sqrt{2}}\frac{4}{\sqrt{3}}:\log_9\frac{1}{4}\)

Answer 1

Ta có :

\(2\log_45=\log_25\)

\(\log_{\sqrt{2}}\frac{4}{\sqrt{3}}=\log_2\frac{4}{\sqrt{3}}=\log_2\frac{16}{3}\)

\(\log_9\frac{1}{4}=\log_{3^2}\left(\frac{1}{2}\right)^2=\log_3\frac{1}{2}\)

Mà :

\(\begin{cases}\frac{1}{2}< \frac{\pi}{4}\Rightarrow\log_3\frac{1}{2}< \log_3\frac{\pi}{4}\\\log_3\frac{\pi}{4}< 0< \log_25\\5< \frac{16}{3}\Rightarrow\log_25< \log_2\frac{16}{3}\end{cases}\)  \(\Rightarrow\log_3\frac{1}{2}< \log_3\frac{\pi}{4}< \log_25< \log_2\frac{16}{3}\)

Hay : 

\(\log_9\frac{1}{4}< \log_3\frac{\pi}{4}< 2\log_45< \log_{\sqrt{2}}\frac{4}{\sqrt{3}}\)

Vậy thứ tự giảm dần là :

\(\log_{\sqrt{2}}\frac{4}{\sqrt{3}};2\log_45;\log_3\frac{\pi}{4};\log_9\frac{1}{4}\)