  In this paper, we study $\mathbb{A}^1$-connected varieties from log geometry
point of view, and prove a criterion for $\mathbb{A}^1$-connectedness. As
applications, we provide many interesting examples of $\mathbb{A}^1$-connected
varieties in the case of complements of ample divisors, and the case of
homogeneous spaces. We also obtain a logarithmic version of Hartshorne's
conjecture characterizing projective spaces and affine spaces.
