Exponential function and natural units The argument of the exponential function has to be dimensionless. By switching to natural units, velocity (for example) becomes dimensionless. Surely, I cannot take the exponential of a velocity now and give it a physical sense, right? However, how do I resolve this contradiction?
 A: 
Surely, I cannot take the exponential of a velocity now and give it a physical sense, right?

Your assumption is incorrect.
In hyperbolic motion under constant proper acceleration $\alpha$, the distance traveled after proper time $\tau$ is $x=\alpha^{-1}(\cosh{\alpha\tau}-1)$ in natural units where $c=1$. Hyperbolic cosines can be written in terms of exponentials, and $\alpha\tau$ is a dimensionless velocity-like quantity (an acceleration multiplied by a time).
A: If you are working in natural units and write an expression like $\exp(v)$, that really means $\exp(v/c)$. Having rescaled the velocity $v$ by the speed of light $c$, in these units, what we write as $v$ is actually the ratio $v/c$ that is often known as $\beta$ is discussions of relativity.  Note that $\beta$, being a ratio of two speeds, is a pure number.
You can write expressions like this one for the Lorentz factor,
$$\gamma=\frac{1}{\sqrt{1-v^{2}}},$$
and that's fine, because in these dimensionless units, $v$ is a quantity that lies in the range $0\leq v<1$.
