1008

已知 ${\displaystyle a={\displaystyle \frac{1}{2}+\ln 2,b=\frac{2}{3}+\frac{\ln 3}{2},c=\frac{1}{2}+\frac{2\ln 5}{5}}}$, 则 ${\displaystyle a,b,c}$ 的大小关系是

作差得$$\begin{array}{rl}& a-c=\frac{5\ln 2-2\ln 5}{5}=\frac{\ln 32-\ln 25}{5}>0\Rightarrow a>c,\\ & b-a=\frac{1+3\ln 3-6\ln 2}{6}=\frac{1}{6}\ln \frac{27\mathrm{e}}{32}>0\Rightarrow b>a,\end{array}$$
故 ${\displaystyle b>a>c}$.