https://www.desmos.com/calculator/m1kaifseq0
in order to get the control points to line up properly, I had to use trial and error to determine the value for this strange constant $k$ at the bottom of my formula sheet. It's approximately $2.299113817$. I have no idea what the arithmetic representation of it is. If anyone can provide some insight to give me some peace of mind, I would greatly appreciate it.
Analytically, $k$ ought to be the value such that $$c+\frac{a\left(c-b\right)}{\ln2}\frac{2^{-\frac{k}{a}}-1}{k} =\frac{c+b}{2}$$
for any arbitrary $a$, $b$, and $c$.
Edit: I've simplified the equation down somewhat to remove the unnecessary constants, and am left with this:
$$\frac{1}{k2^{k}}-\frac{1}{k}=-\frac{\ln2}{2}$$
I'm not sure if this is easier to solve or not
Further edit: according to Wolfram Alpha, the solution is
$$\frac{2+W\left(-\frac{2}{e^{2}}\right)}{\ln2}$$
which is indeed not pretty...
Final edit: I have decided to define the constant like so in my graph:
$$k=\frac{2+\sum_{m=1}^{66}\frac{\left(-m\right)^{m-1}}{m!}\left(-\frac{2}{e^{2}}\right)^{m}}{\ln2}$$
its not pretty, but it is accurate...