Multiply both sides by (2-sqrt(3))^x. This expression is strictly positive, so it will not introduce any new roots.
This yields 1 + (2-sqrt(3))^x = 2^x.
The left side is a strictly decreasing expression and the right side is a strictly increasing function, which means they have at most one intersection point.
Evaluating at x=0 yields left=2 and right=1, so at x=0 the left side is larger. Then evaluating at x=1 yields left=3-sqrt(3)=1.268 and right=4, so at x=1 the right side is larger. Then by the intermediate value theorem there is at least one solution in the interval (0,1).
But since we know there is at most one solution total, then there must be exactly one solution and that solution is on the interval (0,1).
I don't know of any tricks to solve this analytically, but a numeric answer can be calculated to find x~=0.562446.
Solution
