1. In the set of real numbers, solve the equation

$$
\log _{1997}\left(\sqrt{1+x^{2}}+x\right)=\log _{1996}\left(\sqrt{1+x^{2}}-x\right)
$$
Solution. Let $x>0$. Then $\sqrt{x^{2}+1}+x>1$, so $\log \left(\sqrt{x^{2}+1}+x\right)>0$. On the other hand, $\sqrt{x^{2}+1}-x=\frac{1}{\sqrt{x^{2}+1}+x}>0$, which is a solution to the given equation. Similarly, it can be shown that there is no real number $x<0$ that is a solution to the given equation. By direct verification, we find that the only solution to the equation is $x=0$.