Richstone has pioneered the use of the scale-free logarithmic potential for the study of the dynamics of galactic halos. Now modern computers make it practical to exploit that potential in its exact form without central softening. The orbital structure of the potential, which differs significantly from that of the separable Stackel potentials, has been explored systematically with the help of 2-D start spaces. Six triaxial nonrotating density models representative of various axis ratios have been selected. For each model 600 orbits have been integrated over a time interval equivalent to about 55 orbital periods, or 1 Hubble time. Of these orbits, half are defined in the x-z start space (mostly S-tubes, O-tubes, and I-tubes) and the other half in the stationary start space (mostly boxlets such as bananas but also many unstable stochastic orbits). From these orbits self-consistent equilibrium models have been constructed by the usual methods, with the help of a 48 cell grid on the reference sphere and by employing long sequences of Lucy iterations. The main result of these numerical constructions is that in all six cases self-consistent solutions exist. For the rounder models (with density axis ratios c/a >= 0.5) solutions involving only stable orbits exist, although solutions also involving stochastic orbits may be preferred by the halo formation processes. For the flatter models (with density axis ratios c/a ~ 0.3) stochastic orbits have to be included in the solution-a consequence of the replacement of box orbits by boxlets in halo potentials. The inclusion of unstable stochastic orbits in self-consistent equilibrium models raises the question of the persistence of these models. The rate of such impersistence has been estimated and found not threatening under the conditions prevailing in halos where only about 50 orbital periods occur in a Hubble time. It is concluded that equilibrium considerations alone, without the consideration of possible instabilities and without the consideration of halo formation processes, do not suffice to constrain the overall shape of galaxy halos.