I am struggling to understand how exactly the log-sum-exp trick is applied in mixture models (say Gaussian mixture models) and the EM algorithm.
It is well known that parameter estimation for a mixture: $$ f(x)= \sum_{i=1}^K \alpha_i \mathcal{N}(\mu_i,\Sigma_i) = \sum_{i=1}^K \alpha_ie^{-(x-\mu_i)^T\,\, \Sigma_i^{-1} \,\, (x-\mu_i)} $$ can be achieved with the EM algorithm. The E-step corresponds in estimating in which distribution $i$ a given $x$ comes from. This probability is given by: $$ p(x\in \mathcal{N(\mu_i,\Sigma_i)}) = \arg\max_i \frac{\alpha_i e^{-(x-\mu_i)^T\,\, \Sigma_i^{-1} \,\, (x-\mu_i)} }{\sum_{j=1}^{K}\alpha_k e^{-(x-\mu_j)^T\,\, \Sigma_j^{-1} \,\, (x-\mu_j)}} \quad (1) $$ At this point it is common to take the logarithm: $$ (1) \equiv \arg \max_i \Big\{ \log a_i - \log[(x-\mu_j)^T\,\, \Sigma_j^{-1} \,\, (x-\mu_j)] - 1 + \log \sum_{i=1}^N [(x-\mu_i)^T\,\, \Sigma_j^{-1} \,\, (x-\mu_j)] \Big\} $$ where I used the fact that $\log \sum_{i=1}^{N}a_k=1$. I don't understand how the log sum trick is used now in order to conclude the computation. My understanding is that arg min will be replaced by min or similar. At this point I cannot figure it out.