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.
 A: Motivation: "Falls off logarithmically to the right" and "falls off exponentially to the right" means that it increases exponentially in one direction and decreases exponentially in the other.
This sounds like a bit like a logistic function, which models exponential growth as we approach the lower $y$-bound and upside-down exponential growth near the upper $y$-bound (like $y=e^x$ and $y=-e^{-x}$ respectively), but it's rotated inconveniently.
So I looked up its inverse (the logit function), which has the form $\log(\frac{x}{1-x})$. After some modifications to reflect across the $x$-axis and $y$-axis, I got the plot:
$$f(x)=-\log{\frac{|x|}{1-|x|}}$$
(Try $f(ax), af(x), f(x)+a,$ etc. to scale/translate as needed)
A: For instance:
$\arccos(|x| -1)$
Plot
A: How about $\frac{x^2-1}{x^2(x-2)(x+2)}$
A: 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
