I am trying to get my head around the definition of this function (that I concocted as an exercise in defining a function). Let $f$ denote the function satisfying:
$f(0) = +\infty$, and $f(+\infty) = 0$ and is a logarithmic curve.
I know this should be elementary to define but I am stumped.
It's not clear what you mean by "logarithmic curve," but one interesting question is to look for functions of the form $$f(x) = -\log(g(x))$$ that satisfy your properties. Do there exist such $g(x)$?
You want $$+\infty = -\log(g(0))$$ so $g(0) = e^{-\infty} = 0$. You also want $0 = -\log(g(\infty))$ so $g(\infty) = 1$. (And of course, the above equations involving $\infty$ are really just a shorthand for taking limits.)
Finally, while you didn't say so in your question, presumably you want $f$ to decrease monotonically and so $g$ must increase monotonically.
The simplest $g$ would be a polynomial, but you can check that no polynomial can possibly exist with $g(0) = 0, g(\infty) = 1.$ What about rational functions? Here there are many possible solutions, for instance $g(x) = \frac{x}{x+1}$. What about exponentials? $g(x) = a^{-1/x}$ will do the trick for any $a>1$... but here we're cheating, because composing this exponential with $\log$ gives you: $$f(x) = -\log g(x) = -\log a^{-1/x} = -\frac{\log_a a^{-1/x}}{\log_a e} = -\frac{\frac{-1}{x}}{\frac{\log e}{\log a}} = \frac{\log a}{x}$$ which I wouldn't really call "logarithmic curve."
