How to prove that LDA has a similar form to Logistic Regression for the binary classification example? I was studying the book "An Introduction to Statistical Learning" by Gareth James, Daniela Witten, et al. and saw the following line on page 151:

In the LDA framework, we can see from (4.12) to (4.13) (and a bit of
simple algebra) that the log odds is given by:

$$ log \left(\frac{p_1(x)}{1-p_1(x)}\right) = log \left(\frac{p_1(x)}{p_2(x)}\right) = c_0 + c_1(x) $$
How can this proved?

You can view the logistic function as originating from two normal distributed samples.
Say $U \sim N(\mu_U , \sigma)$ and $V \sim N(\mu_Y, \sigma)$ then the relative density of $U$ and $V$ as function of the position is
$$\frac{f_U(x)}{f_V(x)} = \frac{e^{\frac{(x-\mu_U)^2}{2\sigma^2}}}{e^{\frac{(x-\mu_V)^2}{2\sigma^2}}} = e^{\frac{(x-\mu_U)^2 - (x-\mu_V)^2}{2\sigma^2}} = e^{\frac{\mu_U^2+\mu_V^2 - 2x(\mu_U-\mu_V)}{2\sigma^2}} = e^{a+bx}$$
with $a= \frac{\mu_U^2+\mu_V^2}{2\sigma}$ and $b= \frac{\mu_U-\mu_V}{\sigma}$

With LDA you get these two normal distributed samples. LDA assumes that the samples are normal distributed and with the same covariance matrix. In the multidimensional picture you need an extra step but you get that the density ratio along the factor/component (the linear combination found with LDA) is just like the one dimensional picture above. I'll have to make a good illustration of this but you can see this by the points with equal odds ratios being on straight lines/planes (in particular the classifier for odds ratio equal to one is known).
