"exponential growth", "exponentially"

I guess this is how new words supplant old ones.
One of those nasty shocks like learning in childhood that everyone eventually dies
(ok, intentional hyperbole, expressive of a certain degree of exasperation; do not take the above completely literally)
is when you find out that to some people, the phrase "exponential growth" means, simply, extremely or surprisingly fast growth. Now imagine that you're talking about a population that grows at a rate jointly proportional to the present population size and the amount by which the population falls short of the carrying capacity. The population is NOT growing exponentially when it's growing fastest not anywhere near exponentially. But when it's near the beginning of its growth, and growing much more slowly, then it's growing approximately exponentially. Pointing this out seems sure to cause illiterates to find this contradictory.
In childhood, one goes through the reasoning that makes it clear why exponential functions eventually grow faster than polynomials even if in the short run they grow more slowly. Thereafter one understands what "exponential growth" is.

Somehow , it seems, persons who never go throught that line of reasoning come to absorb the phrase "exponential growth" into their vocabulary, and of course, do not even suspect that they are clueless about what it means.
Then someone says something like "We would need an exponentially larger budget in order to ...", so that the word "exponential" now means "very much larger". A new word that is spelled and pronounced the same way as the old one but means something completely different has appeared. What the new word has to do with exponents (those things written in superscript in math) becomes obscure. The whole concept of exponential growth is about the dependence of one quantity on another, but in the new usage, it's about just one quantity. Mike Hardy