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Fundamental Inequality

Lemma ((The Fundamental Inequality for the Logarithm))

log⁡x≤x−1, ∀x>0\log x\le x-1, \ \forall x\gt0

Cor

log⁡y≥1−1y, ∀y>0\log y\ge1-\frac{1}{y}, \ \forall y>0

Cor

∀x>0\forall x>0: log⁡bx≤1log⁡b(x−1)log⁡bx≥1log⁡b(1−1x) \begin{align*} &\log_bx\le\frac{1}{\log b}(x-1) \\ &\log_bx\ge\frac{1}{\log b}(1-\frac{1}{x}) \end{align*}