The Value of logtan10+logtan20+⋯⋯+logtan890 is

$= \log \tan 10+\log \tan 890 + \log \tan 20+ \log \tan 880+\cdots \cdots +\log \tan 450$  

$= \log [\tan 10 \times \tan 890] + \log [\tan 20 \times \tan 880 ] +\cdots \cdots +\log 1$  

$\because \tan (90-\theta )=cot\theta and \tan 450=1$  

= log 1 + log 1 +.....+log 1 

= 0.