It's not as bad as you make it sound: if you write something like `e^t` where t has units of, say, time, then the right way to understand it is that it's actual

It's not as bad as you make it sound: if you write something like `e^t` where t has units of, say, time, then the right way to understand it is that it's actually `e^t (1/T)`, (1/T) is a conversion factor that is normally hidden. `ln e^t = t` then has a hidden `ln (1/T) = -ln T` factor which you can't do much with on its own, but you can carry it forward in equations and eventually if you exponentiate again, it will go back where it belongs. All of the units work out this way. This can be useful because it allows for changing units after the fact. That `ln T` can become a `ln T(S/S) = ln T/S + ln S` if you want, and `ln T/S` can have a non-zero value that actually matters.