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?

One simple way to see it is the following. Transcendental functions typically have a series representation. Now if the argument is a dimensionful quantity then from the series representation it would mean that we are adding quantities of different dimensions. But this we cant do. We can't add apples with oranges. For dimensionless quantities there is no problem.
But there could ofcourse be more rigorous arguments!
