units - Exponential or logarithm of a dimensionful quantity?

I have a unit measure, say, seconds, $s$. Furthermore let's say I have a dimensionful quantity $r$ that is measure in seconds, $s$. What is the unit measure of $e^r$? ($1/r$ is in $Hz$.)

My question is general, how to find the unit measure of a transformation function $y=f(x)$ where $x$ takes some known unit measure. I give above two functions $f(\cdot)=e^\cdot$ and $f(.)=1/\cdot$.

Answer

The only sensible rule when working with units is, that you can only add together terms which carry the same unit. Say $ [x]=[y] $, then $x+y$ is unit-wise a valid statement. You may also multiply arbitrary units together. Whether that is physically sensible is another question. Obviously you cannot add, e.g meters and seconds, but multiplying to form $m/s$ as a unit for velocity is a valid operation.

From that follows, that the argument of the exponential must not carry a unit, because the exponential is defined as a power series. $$ e^x =\sum_{n=0}^\infty \frac{x^n}{n!}$$ If $x$ were to carry a unit, say meters, one would add (schematically) $m+m^2+m^3+\cdots$, which is nonsenical.

If you encounter an exponential, a sine/cosine, logarithm,... in physics you will find almost always that its argument, which must be dimensionless, is a product of often two conjugate variables. Examples are time and frequency, or distance and momentum.