Equations - Exponential Inequation

Exponential Inequation

Definition and Rules Based on Base Value

An exponential inequation is an inequality that involves exponential expressions. The inequality depends on the value of the base \( a \), which affects the direction of the inequality.

Graphical explanation of Exponential Inequation

Key Identity

\[ a^{A(x)} > a^{B(x)} \]

This compares two exponential expressions with the same positive base.

\[ \begin{aligned} &\text{If } a > 1, \quad A(x) > B(x), \\ &\text{If } 0 < a < 1, \quad A(x) < B(x). \end{aligned} \]

When the base is greater than 1, the inequality direction is maintained. When the base is between 0 and 1, the inequality reverses.

Applications

  • Used to compare growth and decay in exponential functions.
  • Common in logarithmic inequality derivations.
  • Helps solve real-world problems involving population growth, compound interest, and radioactive decay.
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