Comment by rq1

The important properties of the logarithm are structural: we usually do not care about units or bases, except when carrying out an actual numerical computation.

As developed in the article, informally, but somewhat sufficiently, the change of base formula shows that the choice of base is largely irrelevant: different bases give equivalent logarithms up to a constant factor.

The Taylor expansion of exp gives a more intrinsic and general definition of the exponential function. This allows exp to be generalised structurally to many algebraic settings, provided the relevant convergence conditions are met: for example, the complex exponential and its many possible logs, the matrix exponential, and so on…