According to the relevant wiki article, there are three types of product integrals. Type II, the geometric integral, is named as the continuous analogue of the discrete product operator.
The geometric integral is defined as follows:
$$\prod_{a}^{b} f(x)^{dx} = \lim_{\Delta x \to 0} f(x_{i})^{\Delta x} = \exp \left( \int_{a}^{b} \ln f(x) \ dx \right). $$
I am looking for some intuition and a motivation behind this definition. It is not clear to me exactly how the equality on the right hand is arrived at, and why it represents a continuous analogue of the discrete product operator.
How can this definition be motivated? And what is the intuition behind it?