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  "body": "\n## Statement\n\nFor every real $x \u003e 0$,\n$$\\ln x \\le x - 1,$$\nwith equality if and only if $x = 1$.\n\n## Remarks\n\nPut $\\ln x$ into the inequality [[$1 + y \\le e^y$|012C7GFF20]]. Then $1 + \\ln x \\le e^{\\ln x} = x$.\nThe line $y = x - 1$ is the tangent to the graph of $\\ln$ at $x = 1$.\nReplacing $x$ by $1/x$ gives $\\ln x \\ge 1 - 1/x$. Mathlib states the main claim as `Real.log_le_sub_one_of_pos`.\n\n## Explanations\n\n### L1 human\n\nThe graph of $\\ln$ lies below its tangent line at $x = 1$.\nThis follows from the concavity of the logarithm.\n",
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