How would one find a real value for $x$ that satisfies
$\tau=\log_x\left(\frac{x+1}{2}\right),$
given $0 < \tau < 1$ and $\tau \neq \frac{1}{2}$
(PS I'm not that good with math, so if this is impossible, please explain it to me like I'm 5). I don't need an exact solution, just a method of estimating one. Thanks!
Context:
I'm trying to create a nonlinear slider for the UI in a program I'm writing. I want to be able to specify the logarithmic scale by specifying the what the output value should be at the midpoint. If I map all the values into a (0,1) range, I need to find an invertible function $f(y)$ such that $0 < \tau < 1$, $f(0) = 0$, $f(1) = 1$, $f(\frac{1}{2}) = \tau$, and $f'(y) > 0$ for all $0 < y < 1$, which also has a nice curve so it doesn't seem jumpy to the user. I realized one such function is:
$f(y) = \log_\alpha\left(1+y\left(\alpha-1\right)\right)$
Now I just need to find the base $\alpha$ given the midpoint $\tau$, which is where all this comes in. I'm not 100% sure my reasoning on this is correct, but it seems like a plausible starting point.