We report an exact analytic representation of the nonlinear Poisson-Boltzmann (PB) potential as a function of radial distance from a cylindrical or spherical polyion in solutions containing a symmetrical electrolyte, in the form of an asymptotic series in elementary functions, generally valid at radial coordinates larger than Debye length. At sufficiently high salt concentrations, where the ratio of Debye length (kappa(-1)) to the polyion radius (a) is sufficiently small ((kappa a)(-1) less than or equal to 1), the asymptotic series is valid at any distance from the polyion surface. This analytic representation satisfies exactly the complete nonlinear Poisson-Boltzmann equation, subject to the boundary condition on the derivative of potential at infinity, and therefore contains one integration constant, which in this salt range we determine to an accuracy of order (kappa a)(-2). Because it explicitly introduces for the first time all the terms which arise due to nonlinearity of the PB equation, this analytic representation clarifies the connection between the exact solution of the PB equation and various approximations including the Debye-Huckel approximation (the solution of the linearized PB equation). From these considerations we obtain a new approximate solution designated "quasi-planar" and expressed in elementary functions, which we show to be accurate at any distance from the polyion surface at typical experimental salt concentrations (e.g., 0.1 M 1:1 salt concentration for double-stranded DNA, where the PB equation retains its accuracy by comparison to Monte Carlo simulations). We apply our analysis to the calculation of the electrostatic free energy and the salt-polyelectrolyte preferential interaction (Donnan) coefficient (Gamma).