>The problem with this approach is that, since we were already introduced to exponents and logarithms in algebra but via different definitions, it always left this unanswered question in my head about how we knew these two definitions were the same, since everyone quickly glossed over that fact.

They just shouldn't be taught in algebra. There, one is thinking about oh how do I extend the definition of exponentiation from integers to real numbers. But thinking about continuous extensions of an integer function to the real line is really something that should be saved for calculus.