We show that similarly to the logarithmic mean-velocity profile in wall-bounded turbulence, the landscape topography presents an intermediate region with a logarithmic mean-elevation profile. Such profiles are present in complex topographies with channel branching and fractal river networks resulting from model simulation, controlled laboratory experiments, and natural landscapes. Dimensional and self-similarity arguments are used to corroborate this finding. We also tested the presence of logarithmic profiles in discrete, minimalist models of networks obtained from optimality principles (optimal channel networks) and directed percolation. The emergence of self-similar scaling appears as a robust outcome in dynamically different, but spatially bounded, complex systems, as a dimensional consequence of length-scale independence.