Equivalence of Logarithm Definitions 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.

I guess this is worth writing down. By your definition we have $\text{exp}(\log(x)) = x$. Differentiating gives
$$\text{exp}(\log(x)) \log'(x) = x \log'(x) = 1$$
hence $\log'(x) = \frac{1}{x}$. The desired result then follows by the fundamental theorem of calculus and the observation that $\log(1) = 0$. 
