Dielectric loss spectra of two glass-forming isomers, eugenol and isoeugenol, measured at ambient and elevated pressures in the normal liquid, supercooled, and glassy states are presented. The isomeric chemical compounds studied differ only by the location of the double bond in the alkyl chain. Above the glass transition temperature T(g), the dielectric loss spectra of both isomers exhibit an excess wing on the high frequency flank of the loss peak of the alpha relaxation and an additional faster gamma process at the megahertz frequency range. By decreasing temperature below T(g) at ambient pressure or by elevating pressure above P(g), the glass transition pressure, at constant temperature, the excess wing of isoeugenol shifts to lower frequencies and is transformed into a secondary beta-loss peak, while in eugenol it becomes a shoulder. These spectral features enable the beta-relaxation time tau(beta) to be determined in the glassy state. These changes indicate that the excess wings in isoeugenol and eugenol are similar and both are secondary beta relaxations that are not resolved in the liquid state. While in both isoeugenol and eugenol the loss peak of the beta relaxation in the glassy state and the corresponding excess wing in the liquid state shifts to lower frequencies on elevating pressure, the locations of their gamma relaxation show little change with increasing pressure. The different pressure sensitivities of the excess wing and gamma relaxation are further demonstrated by the nearly perfect superposition of the alpha-loss peak together with excess wing from the data taken at ambient pressure and at elevated pressure (and higher temperature so as to have the same alpha-peak frequency), but not the gamma-loss peak in both isoeugenol and eugenol. On physical aging isoeugenol, the beta-loss peak shifts to lower frequencies, but not the gamma relaxation. Basing on these experimental facts, the faster gamma relaxation is a local intramolecular process involving a side group and the slower beta relaxation mimics the structural alpha relaxation in behavior, involves the entire molecule and satisfies the criteria for being the Johari-Goldstein beta relaxation. Analysis and interpretation of the spectra utilizing the coupling model further demonstrate that the excess wings seen in the equilibrium liquid states of these two isomers are their genuine Johari-Goldstein beta relaxation.