Successive Detection Logarithmic Amplifier

let's consider this logarithmic Amplifier:

I do not understand why at the terminal LOG OUTPUT there is a current which is proportional to log(Vinput). The current flowing on each diode is equal proportional to exp(-qVouti/KT), where Vouti is the output voltage of each stage.

Each stage is a limiting amplifier: it has a transfer function like this one: