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  "body": "\n## Definition\n\nLet $a \u003e 0$ and let $x$ be a real number.\nThe power $a^x$ is\n$$a^x = e^{x \\ln a}.$$\n\n## Remarks\n\nFor natural numbers $x = n$, this agrees with the product of $n$ factors $a$. For $x = p/q$, it agrees with $\\sqrt[q]{a^p}$.\nThe usual rules hold: $a^{x + y} = a^x a^y$, $(a^x)^y = a^{x y}$, and $(a b)^x = a^x b^x$.\nFor $a = e$, it gives $e^x = \\exp(x)$, since $\\ln e = 1$. Mathlib defines `Real.rpow` with extra conventions for $a \\le 0$.\n\n## Explanations\n\n### L1 human\n\nTo raise $a$ to a real power $x$, multiply the logarithm of $a$ by $x$ and apply the exponential function.\nThis fills in all powers between the rational ones continuously.\n",
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