I am both a Mechanical and an Electrical engineer ( aka use math in real life every day) and I work every day with systems described by exponential or logarithmic functions.
Just to name a few:
- Charging or discharging of a capacitor
- Any LRC circuit (or any combination thereof)
- Any SMD system (or any combination thereof)
- radioactive decay
- algorithmic efficiency
In other words, if you want to describe a real life you will probably encounter some exponential function. This comes from the fact that the solution to differential equations ( which govern most of the universe) generally contain an exponential term.