Algebra

Rewrite the equation as (2017/2018)^x + (2017/2018)^x = 1. The left side is strictly decreasing in x, so there cannot be more than one solution. On the other hand, the left side equals 2 > 1 when x = 0 and goes to 0 when x is very large, so it must equal 1 somewhere in between. Therefore, there is one solution.