I believe SpaceX will only achieve low Earth orbit.
A spacecraft on orbit around the Earth can also be sent to the moon. You just send up more fuel for its rocket motor. Such a spacecraft need only add a little over 4 km/sec of velocity to its orbital speed of 11.86 km/sec to break away from our planet and travel in a free trajectory to the moon.
In any sort of rocketry, getting from Earth’s surface to space is the hard part. This is why rockets use heavy, high-thrust chemically-fueled engines to lift off and boost into orbit. Once you are in space, however, high thrust only reduces travel time; it is no longer a necessity for navigation. An amount of thrust equivalent to that produced by a butterfly’s wing applied over time is enough to get a million-ton spacecraft to Saturn. As the late Robert Heinlein once said, “An object on orbit is halfway to anywhere.”
And what makes you think that SpaceX is bound only for LEO when their goal is to get to Mars?