Why is $\int f(x)^{dx}$ an appropriate notation for the product integral? Just as the title says, why is the product integral's notation: $\int f(x)^{dx}$ appropriate? I sort of understand the product integral, but this notation doesn't really make any sense to me.
Is this a consequence of Theorem $1$:

Theorem 1 If $a_k>−1$ for all $k$, then the infinite product $\prod\limits_{k=1}^\infty (1+a_k)$ converges if and only if the infinite series $\sum\limits_{k=1}^\infty \log{(1+a_k)}$ converges.

If so, how is it a consequence and what is a proof of this? (I found this theorem on How to go from a sum to a product and a product to a sum?, which I found from a google search, so this might be the complete wrong approach at finding out why the notation $\int f(x)^{dx}$ works, but it's the closest I think I have gotten.)
EDIT: So I figure it actually has more to do with how $e^a \cdot e^b = e^{a+b}$ and $\ln{a}+\ln{b} = \ln(ab)$, but how come $e$ and $ln$ don't appear in the integral if you are using this property? Can someone just derive the notation using reimann sums or something?
 A: Wikipedia has a discussion of this.  That article (somewhat unusually) uses the symbol $\Pi$ instead of $\int$.  Other sources may use the $\int$ symbol or the $\int$ symbol  with a superimposed $\times$ or superimposed $\mathcal P$.
I think this is hardly ever used for scalar-valued functions, though.  As noted, you can do it instead with something like $\exp\left(\int \log f(x)\;dx\right)$.  When it is most often used is when the values do not commute.  For example, matrices.  Then the product integral can no longer be reduced to an ordinary integral using $\exp$ and $\log$.
For example, solution of differential equation
$$
\mathbf{x}'(t) = A(t) \mathbf{x}(t)
$$
where $\mathbf{x}(t)$ is an $n$-vector and $A(t)$ is an $n \times n$ matrix for $t \in [a,b]$.
Reference:
Dollard, John D.; Friedman, Charles N., Product integration with applications to differential equations. Foreword by Felix E. Browder. Appendix by P. R. Masani, Encyclopedia ot Mathematics and its Applications 10. Cambridge: Cambridge University Press (ISBN 978-0-521-17737-5/pbk). xxii, 253 p. (2011). ZBL1217.34018.
