Unit Analysis: Exponential & Logarithmic Formulas

[KFC]

Hi all,
I have a general question about the unit in formula or equation. In some formula like $F=md^2x/dt^2$ or thermal radiation law $P \propto A\cdot T^4$, if we plug in the unit for each quantity, the resulting unit of the output is the resulting algebra of the units. For example

$$[F] = \text{kg}\cdot\text{m}^2/\text{s}^2$$

In this case, we can say the unit for the force if kg.m^2/s^2, but what happens if the formula is not linear, for example, if there is a formula $F = \exp(xy)$. I know this formula might not exist in physical world but if it happens to have that and if x and y is not dimensionless, does it mean the unit for F will be exponential? If not, why is that? Why the linear formula will give resulting unit proportional to the individual unit but when the formula becomes nonlinear, they won't give the resulting unit the same way?

Ok, I know that it doesn't have unit like exp(m/t). So does it mean whenever I have formula in exponential or logarithm, the resulting quantity must be dimensionless?

 

[e.bar.goum]

So does it mean whenever I have formula in exponential or logarithm, the resulting quantity must be dimensionless?

The quantity inside the exponential or the logarithm must be dimensionless, yes.

 

[ehild]

Ok, I know that it doesn't have unit like exp(m/t). So does it mean whenever I have formula in exponential or logarithm, the resulting quantity must be dimensionless?

Yes, the exponent must be dimensionless, but such formulae can be of the form Cexp(something) where C has got the appropriate dimension.