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?

Without that, it wouldn't be possible to handle a change of unit with mere factors.

For instance, $e^{2.54 x}$ can't be expressed in terms of $fe^x$ where $f$ would be a suitable conversion factor.

This is by contrast with a power function like $x^3$, such that $(2.54x)^3=(2.54)^3x^3$.