$$H(p,q)=\operatorname {E}_{p}[-\log q]=H(p)+D_{{{\mathrm {KL}}}}(p\|q),\!$$
For discrete $p$ and $q$ this means:
$$H(p,q)=-\sum _{x}p(x)\,\log q(x).\!$$
Logistic loss in the logistic regression is sometimes called cross-entropy loss, which measures the similarity between the prection and actual data labels:
$${\begin{aligned}L({\mathbf {w}})\ &=\ {\frac 1N}\sum _{{n=1}}^{N}H(p_{n},q_{n})\ =\ -{\frac 1N}\sum _{{n=1}}^{N}\ {\bigg [}y_{n}\log {\hat y}_{n}+(1-y_{n})\log(1-{\hat y}_{n}){\bigg ]}\,,\end{aligned}}$$
Because the probability of the data label $y_i$ is 0 or 1 and is fixed, so in the softmax regression, the cross-entropy loss is expressed as:
$$\begin{align} J(\theta) = - \left[ \sum_{i=1}^{m} \sum_{k=1}^{K} 1\left\{y^{(i)} = k\right\} \log \frac{\exp(\theta^{(k)\top} x^{(i)})}{\sum_{j=1}^K \exp(\theta^{(j)\top} x^{(i)})}\right] \end{align}$$

reference

https://en.wikipedia.org/wiki/Cross_entropy
http://ufldl.stanford.edu/tutorial/supervised/SoftmaxRegression/