A function that increases faster than $\ln(x)$ when $x$ is small and then slower than $\ln(x)$ when $x$ is big.

Question

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:

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

Answer 1

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

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.

Answer 2

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 $a\sim 5$.