How could a logarithmically changing graph be made that exists only between asymptotes at x= 0 and 1?

Question

I came up with this problem as a simple thought experiment, but quickly found that it would not be so easy to solve. I managed to come up with these four separate functions that showed some of the properties I wanted, put together making the whole graph. I noticed that each function is very similar to the others, and am convinced that there must be a way to write the four functions as one. $$ f(x) = \begin{cases} -10^{\log_2 (0.5/x)-1}, & 0\le x\le 0.25 \\ -10^{\log_2 (-8(x-0.5))-1}, & 0.25\le x\le 0.5 \\ 10^{\log_2 (8(x-0.5))-1}, & 0.5\le x\le 0.75 \\ 10^{\log_2 (-0.5/(x-1))-1}, & 0.75\le x\le 1 \end{cases} $$ The only parameters are, there must be vertical asymptotes at $x=0$ and $x=1,$ when $x=0.5,$ $y=0,$ the $y$ value must increase/decrease multiplying by ten each time the x value becomes $50\%$ closer to $1$ and $0$ respectively. It should also approach $(0.5,0)$ from both directions in a similar manner.

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Answer 1

It sounds like you're looking for the so-called "logit" function, which is the inverse of a Sigmoidal function. This family of functions is very useful for statistical and machine learning applications. One example is $f(x) = \log\frac{x}{1-x}$, and of course more functions can be obtained simply by scaling to $f(x) = \log\frac{ax}{1-x}$.