We provide exact large-time equivalents of the density and upper tail distributions of the exponential functional of a subordinator in terms of its Laplace exponents. This improves previous results on the logarithmic asymptotic behaviour of the upper tail. Since the results in their general form remain rather abstract, we study some fairly common situations in details to see how the behavior near 0 of the L\'evy measure of the subordinator influences the asymptotics of the density and upper tail distributions. Several examples of applications are then discussed.