What function looks like $\overbrace{}$? Here's my awesome drawing:

Basically it's a function that takes a high (or infinite) value at $0$, then falls off logarithmically for a while before falling off exponentially. 
It doesn't need to be symmetric, it would be ok if the negative $x$ values were 
reversed in sign or something like that. I really only care about $x\ge0$.
Edit: I would prefer if the function is not periodic.

You almost literally describe a function satisfying your description: e.g.
$$ f(x) = -(\ln(x) + \exp(x)) $$
Tweak the constants to taste. Absolute value to make it symmetric: i.e. play with $A,B,C,D$ to shape it how you like:
$$ f(x) = A - B \ln |x| - C \exp(D |x|) $$
Here's a sample function with $A=20$, $B=5$, $C=1/20$ and $d=20$.

generated by wolfram alpha
