Rewrite the equation as $(2003 / 2005)^{x}+(2004 / 2005)^{x}=1$. 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.