The efficient transport of membrane proteins is vital in maintaining life. In this work, we investigate the transport of such membrane proteins along long thin membrane tubes or tethers. We calculate the diffusion constant to leading order in the low Reynolds number regime to be D = (4 pi eta)-1 log(r/a), with r and a being the tube and protein radii, respectively, and eta being the membrane viscosity. Thus we propose an exact limiting form for the controversial logarithmic correction, such as originally introduced by Saffman and Delbruck, that involves the tube radius rather than some "frame size". Our work suggests a test of this logarithmic correction could be achieved by measuring diffusion on membrane tubes, exploiting the fact that the equilibrium tube radius can be controlled by the membrane tension and varied over several orders of magnitude. We analyze the time taken for a protein to transit a membrane tube between cells and find that this can vary by an order of magnitude over physiological tensions. This is a strong effect in biological terms and suggests a possible regulatory coupling between membrane tension and signaling.