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Prerequisites
The logarithm $$\log_a x$$ is defined only when $$x > 0$$, $$a > 0$$ and $$a \neq 1$$.
Before solving any logarithmic inequality, find the domain of every logarithmic expression first. Solve the inequality, then take the intersection of the solution with the domain.
Monotonicity: The core idea
The direction of a logarithmic inequality depends entirely on the base.
If $$a > 1$$, then $$\log_a x$$ is an increasing function:
$$x_1 < x_2 \iff \log_a x_1 < \log_a x_2$$
The inequality sign is preserved.
If $$0 < a < 1$$, then $$\log_a x$$ is a decreasing function:
$$x_1 < x_2 \iff \log_a x_1 > \log_a x_2$$
The inequality sign is reversed.
