Beta Function Quintessence Cosmological Parameters and Fundamental
  Constants II: Exponential and Logarithmic Dark Energy Potentials
  This paper uses the beta function formalism to extend the analysis of
quintessence cosmological parameters to the logarithmic and exponential dark
energy potentials. The previous paper (Thompson 2018) demonstrated the
formalism using power and inverse power potentials. The essentially identical
evolution of the Hubble parameter for all of the quintessence cases and
LambdaCDM is attributed to the flatness of the quintessence dark energy
potentials in the dark energy dominated era. The Hubble parameter is therefore
incapable of discriminating between static and dynamic dark energy. Unlike the
other three potentials considered in the two papers the logarithmic dark energy
potential requires a numerical integration in the formula for the
superpotential rather than being an analytic function. The dark energy equation
of state and the fundamental constants continue to be good discriminators
between static and dynamical dark energy. A new analysis of quintessence with
all four of the potentials relative the swampland conjectures indicates that
the conjecture on the change in the scalar field is satisfied but that the
conjecture on the change of the potential is not.
