[tex]log_2_0_1_8\frac{1}{2}+log_2_0_1_8\frac{2}{3}+log_2_0_1_8\frac{3}{4}+...+log_2_0_1_8\frac{2017}{2018}=[/tex]
[tex]log_2_0_1_8(\frac{1}{2}*\frac{2}{3}*\frac{3}{4}*...*\frac{2016}{2017}*\frac{2017}{2018})=[/tex]
[tex]log_2_0_1_8\frac{1}{2018}=log_2_0_1_8(2018)^{-1}=-1*log_2_0_1_82018=-1*1=-1[/tex]