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Addendum: It has been brought to my attention that the argument herein is specious.

See DOI: 10.1021/ed1000476 "Can One Take the Logarithm or the Sine of a Dimensioned Quantity or a Unit? Dimensional Analysis Involving Transcendental Functions"


Another way to look at it

$$ e^x = \sum_n \frac{x^{n}}{n!} = 1 + x +\frac{x^2}{2} + \dots $$

which comes down to adding quantities with different dimension, which you have already accepted makes no sense. This is why you can't exponentiate values with units.

And we can do a similar thing with most transcendental functions.