Logarithmic finite-size scaling of the O(n)universality class at the upper critical dimensionality(dc = 4)has a fundamental role in statistical and condensed-matter physics and important applications in various experimental systems.Here,we address this long-standing problem in the context of the n-vector model(n = 1,2,3)on periodic four-dimensional hypercubic lattices.We establish an explicit scaling form for the free-energy density,which simultaneously consists of a scaling term for the Gaussian fixed point and another term with multiplicative logarithmic corrections.In particular,we conjecture that the critical two-point correlation g(r,L),with L the linear size,exhibits a two-length behavior:follows r 2-dc governed by the Gaussian fixed point at shorter distances and enters a plateau at larger distances whose height decays as L-dc/2(lnL)(p)with(p)= 1/2 a logarithmic correction exponent.Using extensive Monte Carlo simulations,we provide complementary evidence for the predictions through the finite-size scaling of observables,including the two-point correlation,the magnetic fluctuations at zero and nonzero Fourier modes and the Binder cumulant.Our work sheds light on the formulation of logarithmic finite-size scaling and has practical applications in experimental systems.