13. Find the intersection points of the curves $y=2-\sqrt{1-4^{x}}$ and $y=2^{x} a$ $(a \in \mathbf{R})$.
13. Let $t=2^{x}$.

Then, the problem is transformed into finding the intersection of the line $y=a t$ and the arc of the circle $y=2-\sqrt{1-t^{2}}(00$, and $y \leqslant 2$.
At this point, the intersection point is in the first quadrant, so $02$, the line intersects the arc at one point.
From $a t=2-\sqrt{1-t^{2}}$
$\Rightarrow\left(a^{2}+1\right) t^{2}-4 a t+3=0$
$\Rightarrow t=\frac{2 a \pm \sqrt{a^{2}-3}}{a^{2}+1}$.
In summary, when $a2$, the two curves intersect at one point
$\left(\log _{2} \frac{2 a-\sqrt{a^{2}-3}}{a^{2}+1}, 2-\frac{2+a \sqrt{a^{2}-3}}{a^{2}+1}\right)$.