$\log (\tan 5°)+\log (\tan 25°)+\log (\tan 65°)+\log (\tan 85°)=$

$\log (\tan 5°)+\log (\tan 25°)+\log (\tan 65°)+\log (\tan 85°)$

$=\log (\tan 5°\tan 25°\tan 65°\tan 85°)$

$=\log (\tan 5°\tan 25°\tan (90°-25°)\tan (90°-5°))$

$=\log (\tan 5°\tan 25°\cot 25°\cot 5°)$

$=\log \left((\tan 5°\cot 5°)(\tan 25°\cot 25°)\right)$

$=\log \left(1\right)$

$=0$