Logarithm inequality

For

x > 0

, concavity of the natural logarithm gives

ln x \leq x - 1.

Equivalently,

lo g_{b} x \leq (lo g_{b} e) (x - 1)

for

b > 1

.

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Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 224 / 3 / c / Solution
Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 224 / 4 / d / Solution

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