As discussed in this question, there are many different approaches to defining the natural logarithm function. In particular, since the exponential function$$\exp(x) := \sum_{k=0}^{\infty}\frac{x^k}{k!}$$is strictly increasing, its inverse exists and by definition$$\ln(x) := \exp^{-1}(x).$$On the other hand, then natural logarithm can also be defined through$$\ln(x) := \int_1^{x}\frac{1}{t}dt.$$
What is not at all obvious to me is how these two definitions are equivalent. So, my first question is, how is it that these two definitions are equivalent? A related question is, if one wanted to modify either of these definitions to account for a base other than $e$, how would one proceed? Note that a reference that discusses these topics is perfectly acceptable answer.