How can I prove that the argument of a transcendental function must be dimensionless? We all know from school that arguments of transcendental functions such as exponential, trigonometric and logarithmic functions, or to inhomogeneous polynomials, must be dimensionless quantities. But is there a simple way to prove it?

Analytic expressions $\Psi(x)$, like
$$\Psi(x)={\log(1+x^2)-\cos(3x+5)\over 2\pi(e^{2x}+e^{-5x})}\ ,$$ take "pure" real or complex numbers $x$ as input, and are simply not defined for $x:=$"the distance between earth and moon", or similar.
Dimensional quantities, e.g., $1.5$ volt, $\ 3.75\,\$$, $\ 1{1\over8}$ in, $384\,000$ km, are not elements of ${\mathbb R}$, but elements of some one-dimensional real vector space, and only obtain a numerical value by choosing a basis in that space. This value is then subject to the familiar laws of transformation under change of basis, i.e., of the respective "unit of measure". Of course you are allowed to consider $e^{3.75}$ or $e^{384\,000}$, but these numbers have no meaning whatsoever in connection with the respective problem at hand.
