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  "body": "\n## Statement\n\nFor every real $x \u003e 0$ and every natural number $n$,\n$$\\ln(x^n) = n \\ln x.$$\n\n## Remarks\n\nInduction on $n$ with the [[product rule for logarithms|0K2FNY6DYV]] gives the claim.\nFor [[real exponents|0TV07AQT62]], $\\ln(x^a) = a \\ln x$ holds by the definition $x^a = e^{a \\ln x}$.\nMathlib states it as `Real.log_pow`.\n\n## Explanations\n\n### L1 human\n\nThe logarithm brings an exponent down as a factor.\nFor example, $\\ln(x^3) = 3 \\ln x$.\n",
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