A logarithmic inequation involves logarithmic expressions with a common base. The direction of the inequality depends on the base value \( a \), whether it's greater than 1 or between 0 and 1.
\[ \log_a A(x) = \log_a B(x) \]
Logarithmic inequations compare expressions by taking logs of both sides with the same base.
\[ \begin{aligned} &\text{If } a > 1: \\ &\quad B(x) > 0, \quad A(x) > B(x), \\ &\text{If } 0 < a < 1: \\ &\quad A(x) > 0, \quad A(x) < B(x). \end{aligned} \]
The solution depends on the nature of the base. For bases greater than 1, the inequality is preserved. For fractional bases (0 to 1), the direction reverses.