Units of $e^{-t}$ where $t$ is in seconds ; is there any units?

Question

If you have $y = s^2 e^{-t}$ , what would the units be if $s$ is in meters and $t$ is in seconds, does $y$ even have units? Also what about $y=s^2 \ln(t)$, would you get the same answer ?

Answer 1

Transcendental functions such as $\exp$ and $\ln$ operate on and return pure numbers. So $e^{-t}$ with $t$ in seconds doesn't really make sense; it should be $e^{-ct}$ where $c$ is in $\text{sec}^{-1}$. Perhaps its numerical value happens to be $1\; \text{sec}^{-1}$. And then $s^2 e^{-ct}$ would be in $\text{meters}^2$.