It is common to taking logs of variables when perform a regression analysis. But the observations will become negative after logs transformation when it is less than 1. Is this the possible concerns? (e.g. log(0.5) = -0.6931, but log(5)= 1.6094)
As stated already there is no problem if log(x) is negative. However, you would want to be careful if x=0 or if x<0 because then the logs are undefined. So as long as x>0 then there are no problems at all.
No, $\log(x)$ being negative is not a problem what so ever.