[tex]\it 2^{111}=2^{3\cdot37} =(2^3)^{37}=8^{37}<9^{37}=(3^2)^{37}=3^{74}\ \ \ \ \ (1)\\ \\ 5^{38}=5^{2\cdot19}=(5^2)^{19}=25^{19}<27^{19}=(3^3)^{19}=3^{57}\ \ \ \ \ (2)\\ \\ (1),(2)\Rightarrow 2^{111}\cdot5^{38}<3^{74}\cdot3^{57} \Rightarrow 2^{111}\cdot5^{38}<3^{74+57} \Rightarrow 2^{111}\cdot5^{38}<3^{131}[/tex]
