A lot of the proofs I've found so far are subtly circular because they use properties of the exponential or logarithm function that are derived from the fact (e^x)'=e^x

One good proof I've found defined lnx as the integral of 1/x and worked backwards via partial derivatives before using the proof to derive further properties of e and ln. Is there a more direct method to prove existence than this "I pulled an equation out of my hat (the inverse of the integral of 1/x) and am going to show that it's a solution to my DE thus proving existence"