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

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 f denote the function satisfying:
f ( 0 ) = + , and f ( + ) = 0 and is a logarithmic curve.
I know this should be elementary to define but I am stumped.

Answer & Explanation

kunstdansvo

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
f ( x ) = log ( g ( x ) )
that satisfy your properties. Do there exist such g ( x )?
You want
+ = log ( g ( 0 ) )
so g ( 0 ) = e = 0. You also want 0 = log ( g ( ) ) so g ( ) = 1. (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 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 ( ) = 1. What about rational functions? Here there are many possible solutions, for instance g ( x ) = 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 = log a a 1 / x log a e = 1 x log e log a = log a x
which I wouldn't really call "logarithmic curve."

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