Confusion about cross entropy

Question

I looked at the answer to this question https://stackoverflow.com/questions/41990250/what-is-cross-entropy to try to understand cross-entropy and it seems to me that when the true label for a class is 0, the loss wouldn't increase no matter what the prediction was because the log of the prediction would be multiplied by 0. This doesn't seem like a very accurate way to calculate loss to me. Am I missing something?

Answer 1

$$ \mathcal{L}(\theta)= -\frac{1}{n}\sum_{i=1}^n \left[y_i \log(p_i) + (1-y_i) \log(1-p_i)\right] $$ When $y_i=0$, the second term is nonzero for $p_i \in (0,1)$. The factors involving $y_i$ are like a "switch." Only one of the factors involving $y_i$ is nonzero, so that is the only relevant loss term for sample $i$.