How to define this logarithmic function 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)=+oo, and f(+oo)=0 and is a logarithmic curve. I know this should be elementary to define but I am stumped.
pleitatsj1
Answered question
2022-08-12
How to define this logarithmic function I am trying to get my head around the definition of this function (that I concocted as an exercise in defining a function). Let denote the function satisfying: , and and is a logarithmic curve. I know this should be elementary to define but I am stumped.
Answer & Explanation
kunstdansvo
Beginner2022-08-13Added 16 answers
It's not clear what you mean by "logarithmic curve," but one interesting question is to look for functions of the form
that satisfy your properties. Do there exist such ? You want
so . You also want so . (And of course, the above equations involving are really just a shorthand for taking limits.) Finally, while you didn't say so in your question, presumably you want to decrease monotonically and so must increase monotonically. The simplest would be a polynomial, but you can check that no polynomial can possibly exist with What about rational functions? Here there are many possible solutions, for instance . What about exponentials? will do the trick for any ... but here we're cheating, because composing this exponential with gives you: