Complex geodesics are fundamental constructs for complex analysis and as such constitute one of the most vital research objects within this discipline. In this paper, we formulate a rigorous description, expressed in terms of geometric properties of a domain, of all complex geodesics in a convex tube domain in Cn containing no complex affine lines. Next, we illustrate the obtained result by establishing a set of formulas stipulating a necessary condition for extremal mappings with respect to the Lempert function and the Kobayashi-Royden metric in a large class of bounded, pseudoconvex, complete Reinhardt domains: for all of them in C2 and for those in Cn whose logarithmic image is strictly convex in the geometric sense.