If we accept both positive and negative values for the square root of a number, then can the anti-log of a number be negative?
You should be able to work that out from the definition of the logarithm (and what "antilogarithm" means.)http://en.wikipedia.org/wiki/Logarithm#Inverse_function if ##y=b^x## then ##\log_b(y)=x## ##\text{antilog}_b x (= b^x) = y## You want to know if y can be negative. Presumably your concern is that the log is not defined for negative values of y. It is a bit like the surd for square roots ... to account for negative values, define: ##\log_b|y|=x##, i.e. take the absolute value. Then there are two possible values going the other way. Otherwise you are implicitly requiring a positive value for y as the original input.
Like that - which values we use depends on the context. Maybe we will need both of them. BTW: it is possible to have a negative base ... that can give a negative or a complex antilog.