In the following relationship x and $\delta_S$ are unknowns:
$$\delta_f = \delta_S-(\delta_S-\delta_o)\exp\left[-(1+mx)\frac{It}{V}\right]$$
I need to solve for x and have the following relationship for $\delta_S$:
$$\delta_S =\frac{\delta_{in}+mx\delta^*}{1+mx}$$
Therefore: $$\delta_f = \bigg(\frac{(\delta_{in}+mx\delta^*)}{(1+mx)}\bigg)-\bigg(\frac{(\delta_{in}+mx\delta^*)}{(1+mx)}-\delta_i\bigg)\exp\bigg[-(1+mx)\frac{It}{V}\bigg]$$
Honestly I have no idea where to begin isolating x once I start taking the natural log...
You cannot isolate $x$ and you will need some numerical method.
If I had to do it, I would define $y=(1+mx)$ to make $$\delta_f = \delta_S-(\delta_S-\delta_o)\exp\left[-y\frac{It}{V}\right]$$ $$\delta_S =\frac{\delta_{in}+mx\delta^*}{1+mx}=\frac{\delta_{in}-\delta^*+y\delta^*}{y}=\delta^*+\frac{\delta_{in}-\delta^*}y$$ Use Newton method.
In fact, I suppose that there could be an "explicit" solution in terms of the generalized Lambert function (have a look at equation $(4)$ in the paper).