Example 1. Solve the inequality

$$
\log _{7} \frac{x-2}{x-3}<0
$$
Solution. Since the base of the logarithm is greater than one, the given inequality is equivalent to the double inequality

$$
00 \\
\frac{x-2}{x-3}0
$$

is satisfied for all $x$ from the intervals $-\infty<x<2$ and $3<x<$ $<+\infty$.

The inequality

$$
\frac{x-2}{x-3}<1
$$

is equivalent to the inequality

$$
\frac{x-2}{x-3}-1<0
$$

i.e., the inequality

$$
\frac{1}{x-3}<0
$$

the solution of which is $x<3$.

The intersection of the sets of solutions of each inequality in the system is the interval $-\infty<x<2$, which is the set of all solutions to the original inequality.