Logarithmic inequalities (including rationalization method)

Logarithmic inequalities

A logarithmic inequality is one where the unknown appears inside a logarithm or in bases/exponents of expressions with $\log$ . As with equations, first find the domain: all $\log$ arguments must be $>0$ , bases $>0$ , $\neq1$ . The key step is monotonicity of $\log_a t$ on $(0;+\infty)$ : • if $a>1$ , it strictly increases: $\log_a u<\log_a v\ \Leftrightarrow\ u<v$ (and similarly for $\le,>,\ge$ when $u,v>0$ ); • if $0<a<1$ , it strictly decreases: $\log_a u<\log_a v\ \Leftrightarrow\ u>v$ — the order of the arguments flips. After you reduce to an algebraic inequality in $x$ , intersect with the domain.

Same pattern as $a^x$ : for $a>1$ the sign copies; for $0<a<1$ it flips

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