# Difference between Softmax and multiclass logistic regression?

I know that logistic regression is for binary outputs and Softmax is for multiple classes. Would it be fair to say that Softmax regression is the same thing as multiclass Logistic regression?

Remember in logistic regression labels and model parameters were: $$y^{(i)} \in \{0,1\},\space \theta = \begin{bmatrix} \theta_1 \\ \theta_2 \\ \vdots \\ \theta_n \end{bmatrix}$$ Whereas in softmax regression labels and model parameters are: $$y^{(i)} \in \{1, 2, \ldots, K\},\space \theta = \begin{bmatrix} \theta_1^1 & \theta_1^2 & \theta_1^k \\ \theta_2^1 & \theta_2^2 & \theta_2^k \\ \vdots & \vdots & \vdots\\ \theta_n^1 & \theta_n^2 & \theta_n^k \\ \end{bmatrix}$$ In that sence it is easy to see that logistic regression can be expressed as softmax with two classes. Of course, cost functions and hypothesis are a bit different.