  We study global well-posedness of strong solutions for the nonhomogeneous
Navier-Stokes equations with density-dependent viscosity and initial density
allowing vanish in $\mathbb{R}^2$. Applying a logarithmic interpolation
inequality and delicate energy estimates, we show the global existence of a
unique strong solution provided that $\|\nabla\mu(\rho_0)\|_{L^q}$ is suitably
small, which improves the previous result of Huang and Wang [SIAM J. Math.
Anal. 46, 1771--1788 (2014)] to the whole space case. Moreover, we also derive
exponential decay rates of the solution. In particular, there is no need to
require additional initial compatibility condition despite the presence of
vacuum.
