@joeyadams said:
e^(1 meter), by Taylor expansion, would have to equal 1 unit plus 1 meter plus 1/2 a square meter plus 1/6 a cubic meter plus...
If anyone's building a spaceship in a Hilbert space or something that might be useful :)
It's beginning to dawn on me that most people seem to be victim of a sub-prime science education. Quantities don't "have" units. 'm' in 'exp(17m)' is a mere tag, it's a reminder that that particular number came about by having to lay 17 meter-long rods end-to-end to match the length of the object. We always form exponentials of numbers.
Systems of measurement are arbitrary, but a general statement such as 'area=length*height' should not depend on the system chosen. 'Dimensional analysis' is just that: making sure we're not dealing with statements that are true by mere coincidence by checking their independence of the system of units.
As long as one is dealing with sums, products, and fractions, this can be done by treating the tags on the same footing as positive real constants, multiply them, raise them to powers, and reduce fractions. In the end, one checks for algebraic homogeneity. That works so well in practice that people hypostasize mysterious algebraic objects named "Meter", "Volt" and "Ampere" (stored in an underground Banach space in Paris, I assume).
Now, it's easy to check that log(length/height) = log(length) - log(height) holds independent of the measuring rods one happens to use for length and height. It's just not possible to check by inspecting just one term of the sum (say, log(length)). Well, of course not. Logarithms turn products into sums. But for some reason, this deeply upsets people; much more than the fact that they cannot dimension-check 'area=length*height' by looking at 'height' alone.