Thermodynamics of logarithmic charged black holes in the Einstein-dilaton gravity theory

Starting from the suitable action of four-dimensional Einstein-dilaton gravity with the logarithmic nonlinear electrodynamics, the explicit form of the coupled scalar, electromagnetic and gravitational field equations have been obtained. Through the exact solution of the field equations, the nonvanishing component of the electromagnetic tensor have been obtained and it has been shown that dilaton potential can be written as the suitable combination of two terms similar to the Liouville potentials. Three classes of novel electrically charged hairy black holes have been introduced with the regular and well-defined metric functions for all values of the parameters in the theory. The solutions reduce to their corresponding values in the Einstein-Maxwell-dilaton theory when the nonlinearity parameter of the electrodynamics goes to infinity. The thermodynamic quantities, such as temperature, entropy, mass, electric charge and electric potential, have been calculated and it has been demonstrated that the first law of black hole thermodynamics is valid for all of the new solutions. Thermodynamic stability or phase transition of the black holes have been investigated based on the canonical ensemble method. By calculating the black hole heat capacity type-1 and type-2 phase transition points have been determined. Also, it has been found that in order for the new nonlinearly charged dilaton black holes to remain locally stable some simple conditions must be fulfilled.