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Nope : without further hypotheses, neither Sage, Sympy nor Mathematica can do better than :

sage: constants=var('a1,a2,b,c')
sage: f=((a1*b)^c/(a2*b)^c)
sage: f.canonicalize_radical()

Powers and logs are simple to treat in the "usual" special cases and extremely tricky in general. Hiont : think of lomplex logarithm and its branch cut...