Different logistic regression models

Question

I've been revising the concepts of logistic regression and suddenly realized that the probability function of the logistic model in the book ISL looks absolutely different from other sources. For ISL logistic function looks this way:

$P(x) = \frac{e^{(\beta_0 + \beta_1x)}}{1 + e^{(\beta_0 + \beta_1x)}}$

(sorry for my lack of knowledge of acceptable formatting). For other sources (especially, courses) logistic function looks another way:

$P(x) = \frac{1}{1 + e^{(\beta_0 + \beta_1x)}}$

(so-called sigmoid function). Moreover, 2 different approaches are used to estimate the coefficients $\beta_0$ and $\beta_1$: for the first equation we're maximizing the likelihood function, while for another we're minimizing the cost function for this model.

Can you please explain me the difference (in simple words) and tell what approach should be used for what case?

Answer 1

(Summarizing answer in comments) They are really the same, after you put in that minus sign you lost. Just multiply in numerator and denominator of first form with $\exp(-\beta_0-\beta_1 x)$.

Logistic regression is usually estimated by maximizing the likelihood function, but the negative of the loglikelihood function is often (especially in machine learning circles) called the log-loss. It is really the same thing, only wrapped differently.