  Radiative symmetry breaking provides an appealing explanation for electroweak
symmetry breaking and addresses the hierarchy problem. We present a
comprehensive phenomenological study of this scenario, focusing on its key
feature: the logarithmic-shaped potential. This potential gives rise to a
relatively light scalar boson that mixes with the Higgs boson and leads to
first-order phase transitions (FOPTs) in the early Universe. Our detailed
analysis includes providing exact and analytical solutions for the vacuum
structure and scalar interactions, classifying four patterns of cosmic thermal
history, and calculating the supercooled FOPT dynamics and GWs. By combining
future collider and gravitational wave experiments, we can probe the conformal
symmetry breaking scales up to $10^5-10^8$ GeV.
