Logarithmic Multi - Stage Amplifier
If you take only the output from the final amplifier, then the transfer function is:


At some value of input, the output stage will saturate and not increase any further.

If we then add the other outputs, we will get (not to scale):



As each amplifier saturates, then the slope reduces until the next amplifier in the  chain saturates until all 4 amplifiers are in saturation. As each amplifier output is lower then the following devices the output slope from the addition reduces.

By adding these outputs together we get a log amp shown (not complete and also not to scale) by the purple line.

This is simplistic, but it is the fundamental principle of operation.

The output slope is for a setup where all the amplifiers have a gain of A:

All amplifier still active: 10A  (4+3+2+1)

4th amplifier saturated: 6A (3+2+1)

3rd and 4th amplifier saturated: 3A (2+1)

2nd, 3rd and 4th amplifier saturated: A

So the output slope reduces as the level of input is raised.