The log-likelihood function of Multinomial logistic regression is given by:
\begin{equation} l(w) = \sum_{j=1}^{n}\left(\sum_{i=1}^{m}{y_{j}}^{(i)}{w^{(i)}}^{T}x_{j} - \log\left(\sum_{i=1}^{m}\exp\left({w^{i}}^{T}x_{j}\right)\right)\right) \end{equation}
where $n$ - no. of samples , $m$ - no. of classes. $x_{j}$ - $j^{th}$ training data. We know for $m=2$, Multinomial Logistic Regression reduces to Logistic Regression. By substituting $m=2$ and by assuming weight for a specific class is zero , I get:
\begin{equation} l(w) = \sum_{j=1}^{n} \left({y_{j}}{w}^{T}x_{j}- \log\left(1+\exp\left(w^{T}x_{j}\right)\right)\right) \end{equation}
But the actual log-likelihood function of Logistic Regression is just: \begin{equation} l(w) = -\sum_{j=1}^{n}\log\left((1+\exp\left(w^{T}x_{j}\right)\right) \end{equation}
Where did the first linear term vanish?