Function that represents growth with specific specs

Question

I'm looking for a function that represents growth based on the following specs:

In a range of 365 days the function may grow from (almost) zero to a maximum of 1. It should have some kind of logarithmic shape, because the growth in the first 30 days should happen quite fast (~ 50%).

Anyone an idea how to achieve that? Thanks in advance,

Hitch

EDIT: Made a major mistake: the range is 365 days, not 100!

That's how it should look like:

Answer 1

For passing through $(100,a)$ and $(30, 0.5a)$, I have a function using $\sin$:

$$f(x)=\sin\left(45\cdot \sin \left(x \right) \right).$$