A manifold with a smooth group structure is called a Lie group. Most of the information about Lie groups is captured by the tangent space at the identity and its Poisson bracket. The map relating these two structures is the exponential map (which in the compact case is the same as the geodesic exponential map). In this talk I'll start from first principles, give a brief overview of some decompositions of Lie groups based on this algebraic/analytic interplay, and use these facts to prove some interesting theorems about the image of $\exp$, viz., 1) If $\exp$ is a homeomorphism, then the group is solvable; and 2) while $\exp$ need not be surjective, the connected component of the identity is the square of the image of $\exp$.