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  "body": "\n## Statement\n\nThe [[natural logarithm|0044CPHJBV]] is strictly [[concave|0EF2VG8XKQ]] on $(0, \\infty)$.\nThat is, $\\ln(t x + (1 - t) y) \u003e t \\ln x + (1 - t) \\ln y$ for all $x \\ne y$ in $(0, \\infty)$ and $t \\in (0, 1)$.\n\n## Remarks\n\nThe [[derivative of $\\ln x$|07X1QA6BRW]] is $1/x$, and its second derivative is $-1/x^2 \u003c 0$.\nThe concavity gives the [[AM–GM inequality|0DRF0F3XC9]] through Jensen's inequality.\nMathlib states it as `strictConcaveOn_log_Ioi`.\n\n## Explanations\n\n### L1 human\n\nThe graph of $\\ln x$ bends downward everywhere.\nEvery chord of the graph lies below the graph.\n",
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