Admittedly, All trueSo App Note 311 has a use for an exponential amplifier, and describes a gas guage application that uses an ultrasound pulse to determine fuel level. The anti-log amplifier allows increased resolution as the tank approaches empty. These were for pumps in an irrigation system.
The app note was published in 1991 and DSP has come a long way since then. The other clue to the application's antiquity was the use of 74Cxxx logic at a Vcc of 15 volts. This stuff dates to the 1974 time frame.
This certainly wins my vote for obscurity, but possibly there is a useful pearl in there.