Number of solutions

Relevant wiki: Euler's Formula

$\begin{aligned} (1-i)^x & = 2^x \\ (\sqrt 2)^x {\color{#3D99F6}\left(\frac 1{\sqrt 2} - \frac i{\sqrt 2} \right)^x} & = 2^x & \small {\color{#3D99F6} \text{By Euler's formula}} \\ 2^\frac x2 {\color{#3D99F6}e^{-\frac {\pi x} 4i}} & = 2^x \\ \left| 2^\frac x2 e^{-\frac {\pi x} 4i} \right| & = \left| 2^x\right| \\ 2^\frac x2 \left| e^{-\frac {\pi x} 4i} \right| & = 2^x \\ 2^\frac x2 (1) & = 2^x \\ \implies x & = \boxed{0} \end{aligned}$

$(1-i)^x=2^x$

Of-course, for $x=0$ $1=1$

$(1-I)^x-2^x = 0 \\ \implies 1-i=0, 2 =0$

both which is not possible. Therefore it have only one integral solution