Linear discriminant analysis implies a logistic model
For two normal classes with common covariance matrix \Sigma, means \mu_1, \mu_0 and priors \pi_1, \pi_0, show that \log(\tau_1(x)/\tau_0(x))=\alpha_0+\beta^\top x and identify \alpha_0 and \beta.
Worked answer (click to show)
\log\frac{\tau_1}{\tau_0} =\log\frac{\pi_1}{\pi_0}+\log\frac{f_1(x)}{f_0(x)}.
With the common covariance the quadratic terms cancel:
\begin{align*} \log\frac{f_1}{f_0} &=-\tfrac12(x-\mu_1)^\top\Sigma^{-1}(x-\mu_1) +\tfrac12(x-\mu_0)^\top\Sigma^{-1}(x-\mu_0)\\ &=x^\top\Sigma^{-1}(\mu_1-\mu_0) -\tfrac12(\mu_1+\mu_0)^\top\Sigma^{-1}(\mu_1-\mu_0). \end{align*}
So \beta=\Sigma^{-1}(\mu_1-\mu_0) and
\alpha_0=\log\frac{\pi_1}{\pi_0}-\tfrac12(\mu_1+\mu_0)^\top\Sigma^{-1}(\mu_1-\mu_0).
This is a logistic model with those parameters.