# Why is the logistic regression decision boundary linear in X?

The logistic regression model,

$$\operatorname{p}(X) = \frac{\operatorname{e}^{\beta_0 + \beta_1 X}}{1 + \operatorname{e}^{\beta_0 + \beta_1 X}}$$

is said to create a decision boundary linear in $X$.

As far as I understand, only the logit is linear in $X$. Is this the reason the decision boundary is linear in X? If so, why? And if this is not the case, what is the reason for this phenomenon?

I am confused about this because the decision boundary can be expressed as: $$\operatorname{p}(X) = a, a \in [0,\, 1]$$ And $\operatorname{p}(X)$ is not linear in $X$.

The decision boundary $p(X)=a, a\in(0,1)$ is
$$\beta_0+\beta_1X=\log\frac{a}{1-a}$$
which is linear in $X$.
• Ah, thanks for showing me the relation of the decision boundary to the logit. The decision boundary is thus a hyperplane, and thus linear in $X$. – Nurmister Sep 15 '18 at 16:43