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  "body": "\n## Statement\n\nFor every real $x \u003e 0$, $e^{\\ln x} = x$. For every real $y$, $\\ln(e^y) = y$.\nThe function $\\ln$ is continuous and strictly increasing on $(0, \\infty)$, and its range is $\\mathbb{R}$.\n\n## Remarks\n\nThe two identities say that $\\ln$ and $\\exp$ are inverse bijections between $(0, \\infty)$ and $\\mathbb{R}$.\nContinuity follows from the [[continuity of inverse functions|07BY47HG5S]], since $\\exp$ is [[strictly increasing|0KN2V7840E]] and continuous.\nAlso $\\ln x \\to \\infty$ as $x \\to \\infty$, and $\\ln x \\to -\\infty$ as $x \\to 0^+$. Mathlib states the first identity as `Real.exp_log`.\n\n## Explanations\n\n### L1 human\n\nTaking $\\ln$ undoes taking $\\exp$, and the other way round.\nThe graph of $\\ln$ is the mirror image of the graph of $e^x$ in the line $y = x$.\n",
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