  This paper investigates thermodynamics, quasi-normal modes, thermal
fluctuations and phase transitions of Reissner-Nordstr\"om black hole with the
effects of non-linear electrodynamics. We first compute the expressions for
Hawking temperature, entropy and heat capacity of this black hole and then
obtain a relation between Davies's point and quasi-normal modes with non-linear
electrodynamics. We also observe the effects of logarithmic corrections on
uncorrected thermodynamic quantities such as entropy, Hawking temperature,
Helmholtz free energy, internal energy, Gibbs free energy, enthalpy and heat
capacity. It is found that presence of non-linear electrodynamic parameter
induces more instability in black holes of large radii. Finally, we analyze the
phase transitions of Hawking temperature as well as heat capacity in terms of
entropy for different values of charge ($q$), horizon radius ($r_{+}$) and
coupling parameter ($\alpha$). We obtain that Hawking temperature changes its
phase from positive to negative for increasing values of $q$ and $r_{+}$ while
it shows opposite trend for higher values of $\alpha$. The heat capacity
changes its phase from negative to positive for large values of charge, horizon
radius and coupling parameter.
