A method based on the modulus of curve families is introduced in the Heisenberg group \(H^1 = \mathbb{C} \times \mathbb{R}\). The left invariant metric \(d_H(p,q)\), the Heisenberg distance, is employed to define quasiconformality of a mapping in the usual metric way, see e.g. [A. Koranyi and H. M. Reimann, Adv. Math. 111, No. 1, 1–87 (1995; Zbl 0876.30019)] for equivalent definitions. The conformally invariant 4–modulus \(M_4(\Gamma)\) of a curve family \(\Gamma\) gives another way to define quasiconformality of a mapping in \(H^1\) via the modulus inequalities. These basics concepts are carefully considered in the Appendix. The authors then proceed to consider minimization problems related to the maximal distortion and to the mean distortion. The main purpose of the paper is to use so called logarithmic coordinates in \(H^1\); these were introduced by I. D. Platis [Math. Proc. Camb. Philos. Soc. 147, No. 1, 205–234 (2009; Zbl 1176.53059)] and in these coordinates the coordinate planes are the Heisenberg unit sphere, the complex plane and the standard flat pack. Moreover, the difficult contact conditions are easier to interpret in logarithmic coordinates. Following the analysis on extremal properties of the stretch mapping in [Z. M. Balogh, K. Fässler and I. D. Platis, J. Anal. Math. 113, 265–291 (2011; Zbl 1232.30019)] the authors examine the natural extension of the classical stretch mapping between annuli in \(H^1\). The paper contains useful modulus estimates and a number of open problems related to quasiconformality in \(H^1\).