Exponential inequalities

An exponential inequality has the unknown in the exponent. Often: reduce to one base $a>0$ , $a\neq1$ , then compare exponents. Key fact — monotonicity of $a^x$ : • if $a>1$ , the function strictly increases: $a^{u}<a^{v}\ \Leftrightarrow\ u<v$ , similarly for $\le,>,\ge$ ; • if $0<a<1$ , it strictly decreases: $a^{u}<a^{v}\ \Leftrightarrow\ u>v$ — the inequality between exponents flips. After reducing to an inequality in $x$ , solve it with usual school tools (linear, quadratic, union of intervals).

Always clarify: base greater than 1 or between 0 and 1 — the flip depends on that

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