In an engineering problem I was working on, I have found a function of three variables:
$$f(x, y, k) = k\cdot\Bigg(1-\bigg(\frac{1}{2}\bigg)^{\frac{x}{y}}\Bigg) = x $$
In certain applications, we hold $y$ and $k$ constant, and consider it to act as a function of one variable. In that case, is it possible to express the zeroes of the function $f(x) = 0$ in closed form in terms of $y$ and $k$? Of course the solutions can be found numerically, but it would be convenient to have a closed form if possible.
For example, in one case, $k=134201.63$ and $y=5\cdot10^4$. In this case, the zero occurs at approximately $x=101208.723$.
Let set $X=\frac xk$ and $a=\left(\frac 12\right)^{\frac ky}$
The equation becomes $1-a^X=X$
No set $U=1-X$ then $U=a^X=a^{1-U}=aa^{-U}\iff Ua^U=a$
Finally for $\alpha=\ln(a)$ and $V=\alpha U$ leads to $$Ve^V=a\ln(a)=\alpha e^\alpha$$
But $xe^x$ is not injective, its reciprocal the Lambert $W$ function has two branches $W_0$ and $W_{-1}$
https://en.wikipedia.org/wiki/Lambert_W_function
From the figures you have given we have $\alpha<-1$ so the two solutions are
$V_1=W_{-1}(\alpha e^\alpha)=\alpha\implies U_1=1\implies x_1=0$
$V_2=W_0(a\ln(a))\implies x_2=k\left(1-\dfrac{W_0(a\ln(a))}{\ln(a)}\right)$
For other values of $a$, it may happen that the non-zero value is reached for $W_{-1}$ instead.
You have to evaluate though, whether calculating this thing is more efficient than a numerical solving of the initial equation or even numerically solve the simpler $Ua^U=a$, I guess it all comes to whether you have a good implementation of the LambertW function on your platform or not.