A formula that yields a particular graph shape I would like a formula for a function whose graph has the following properties:

$f(0) = 0$.

$\lim\limits_{x\to\infty}f(x) = y$.

The shape of the function is approximately the following:


It should have an exponential or a logarithm in the formula.


Any function like this?
The graph of $y=5-5e^{-x}$ has the desired characteristics. More generally, if you want $\lim\limits_{x\to\infty}f(x)=a&gt;0$, the function $f(x)=a-ae^{-x}$ works.
Also your graph doesn't have anything shown for x&lt;0, so presumably its equal to zero there?

i.e. Use a piecewise defined function:

f(x) = a-a*exp(-x) : for positive x and f(x)=0 for non-positive x.