Given are two equations:
$$v_1 = v_0 (1 - e^{-\frac{t_1}{\tau}})$$
$$v_2 = v_0 (1 - e^{-\frac{t_2}{\tau}})$$
We know that
$$t_2 > t_1$$ $$v_2 > v_1$$ $$\tau > 0$$ $$v_0 > 0$$ $$\tau, v_0 \in ℝ$$
Given $t_1, v_1, t_2, v_2$, how can we solve for $\tau, v_0$?
Let $p:=e^{-t_1/\tau}$ so that $e^{-t_2/\tau}=p^\alpha$, where $\alpha$ is known. The equation can be written
$$v_1(1-p^\alpha)=v_2(1-p).$$
$\alpha$ can be an integer $>4$ so that there are certainly cases such that there is no closed-form solution. (In fact, there are only closed-form solutions for a few rational values of $\alpha$.)