  In this paper, we study the power-law $f(T)$ model using Hubble diagrams of
type Ia supernovae (SNIa), quasars (QSOs), Gamma Ray Bursts (GRBs) and the
measurements from baryonic acoustic oscillations (BAO) in the framework of the
cosmographic method. Using mock data for SNIa, QSOs and GRBs generated based on
the power-law $f(T)$ model, we show whether different cosmographic methods are
suitable to reconstruct the distance modulus or not. In particular, we
investigate the rational PADE polynomials $(3,2)$ and $(2,2)$ in addition to
the fourth- and fifth- order Taylor series. We show that PADE $(3,2)$ is the
best approximation that can be used in the cosmographic method to reconstruct
the distance modulus at both low and high redshifts. In the context of PADE
$(3,2)$ cosmographic method, we show that the power-law $f(T)$ model is well
consistent with the real observational data from the Hubble diagrams of SNIa,
QSOs and GRBs. Moreover, we find that the combination of the Hubble diagram of
SNIa and the BAO observation leads to better consistency between the
model-independent cosmographic method and the power-law $f(T)$ model. Finally,
our observational constraints on the parameter of the effective equation of
state of DE, described by the power-law $f(T)$ model, show the phantom-like
behavior, especially when the BAO observations are included in our analysis.
