Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

As the title indicate:

I am looking for a function that increases faster than $\ln(x)$ when $x$ is small and then slower than $\ln(x)$ when $x$ is big.

Here is the fig:

enter image description here

The red curve is the $\log(x)$, and the green curve is the function I am looking for.

share|improve this question
    
Do you want the green curve to have a vertical and a horizontal asymptote like the figure suggests? –  Hurkyl Jul 16 at 17:02
    
@Hurkyl Ideally the green curve approaches 1 when x is infinity. –  Leo Jul 16 at 17:07
    
@Leo The error function is one such function. See en.wikipedia.org/wiki/Error_function. Although it is not an elementary function. –  Eff Jul 16 at 17:08
    
I think you should give more specifications: desired range, values and slopes at the ends of the range. –  Yves Daoust Jul 16 at 17:21

2 Answers 2

up vote 1 down vote accepted

What about $\sqrt{\log x\log{10}}$ ?

enter image description here

Actually, any function mapping $0$ to $0$ and $\log10$ to $\log10$ and such that $f(x)>x$ in between can be used to compose $f(\log x)$ like you want.

You can also use Hermite cubic interpolation, giving you all freedom to adjust the slopes.

share|improve this answer

Your requirements "when x is small" and "when x is large" are a bit vague.

However, try the function:

$$f(x)=x^{\frac{a}{x}}$$

and adjust $a>1$ to fine-tune the function's descent. Try with $a\sim 5$.

enter image description here

share|improve this answer
    
I add a figure for the function u mentioned. However, it decreases after some x value. I need a increase function f(x) (when x>0). –  Leo Jul 16 at 17:17
    
@Leo: I thought you said (in above comment) that the green function "ideally" should approach 1, when $x\to\infty$. –  ioannis galidakis Jul 16 at 17:21
    
This bears little resemblance with what the OP asked for. –  Yves Daoust Jul 17 at 16:36
1  
@Yves Daoust: The OP changed his mind, midway after some of the answers were given. His answer to Hurkyl: "Ideally the green curve approaches 1 when x is infinity.", etc. Then, below my answer: "However...I need a increase function $f(x)$ (when $x>0$). –  ioannis galidakis Jul 17 at 17:23
1  
I downvoted him. –  Yves Daoust Jul 17 at 21:53

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.