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  "body": "\n## Definition\n\nFor a real number $x \u003e 0$, the natural logarithm $\\ln x$ is the unique real number $y$ with $e^y = x$.\nThis defines a function $\\ln \\colon (0, \\infty) \\to \\mathbb{R}$.\n\n## Remarks\n\nThe number $y$ exists and is unique, because $\\exp$ is a [[strictly increasing|0KN2V7840E]] continuous bijection from $\\mathbb{R}$ onto $(0, \\infty)$.\nSome texts define instead $\\ln x = \\int_1^x \\frac{dt}{t}$. Both definitions give the [[same function|0K1N1Y4EGZ]].\nOther notations are $\\log x$ and $\\log_e x$. Mathlib defines `Real.log` for all reals, with $\\log x = \\log |x|$ and $\\log 0 = 0$.\n\n## Explanations\n\n### L1 human\n\nThe logarithm answers the question: to which power must we raise $e$ to get $x$?\nFor example, $\\ln 1 = 0$ and $\\ln e = 1$.\n",
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