First, this thing → cannot be equal to zero. The best you can get is for it to tend to zero as z_1 tends to negative infinity.
However, in a computer, it is possible because it is not infinitely precise, and in that case, one usual trick is to add a very small number to make this → log ( 1e-7 + a) such that even if a_i numerically becomes 0, the 1e-7 will take care of that. Below is a relevant post on how Tensorflow uses a small number called epsilon to handle that.
cannot be equal to zero. The best you can get is for it to tend to zero as z_1 tends to negative infinity.