Example 2. If $x, y$ are real numbers, and
$$
y=\frac{\left(1-x^{2}\right)^{\frac{1}{2}}+\left(x^{2}-1\right)^{\frac{1}{6}}}{4 x-5} \text {, }
$$

find the value of $\log _{\frac{1}{7}}(x+y)$.
To make the expression for $y$ meaningful, $x$ must simultaneously satisfy the following three inequalities.
$$
\begin{array}{c}
1-x^{2} \geqslant 0, \\
x^{2}-1 \geqslant 0, \\
4 x-5 \neq 0 . \\
\therefore x=1, \quad \text { thus } y=0 . \\
\therefore \log _{\frac{1}{7}}(x+y)=\log _{\frac{1}{7}} 1=0 .
\end{array}
$$