Logarithmic Inequalities

Definition: Logarithmic Inequality

A (real) logarithmic inequality is an [[index|inequality]] which contains variables as part of the base or the argument of a real logarithm.

Algorithm: Solving Inequalities of the Form \(\log_a f(x) \gt \log_a g(x)\)

We are given the following inequality:

\[ \log_a f(x) \gt \log_a g(x) \]

Solutions:

If \(0 \lt a \lt 1\), then \(\begin{cases}f(x) \lt g(x) \\ f(x) \gt 0\end{cases}\)
If \(a \gt 1\), then \(\begin{cases}f(x) \gt g(x) \\ g(x) \gt 0\end{cases}\)

Algorithm: Solving Inequalities of the Form \(\log_{b(x)}f(x) \ge \log_{b(x)} g(x)\)

We are given the following inequality:

\[ \log_{b(x)}f(x) \ge \log_{b(x)} g(x) \]

Solutions:

\[ \begin{cases}0 \lt b(x) \lt 1 \\ f(x) \le g(x) \\ f(x) \gt 0\end{cases} \qquad \bigcup \qquad \begin{cases}b(x) \gt 1 \\ f(x) \ge g(x) \\ g(x) \gt 0 \end{cases} \]