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  "body": "\n## Summary\n\nCauchy's power equation is the [[functional equation|SRAN4EKWS0]] $f(xy) = f(x) f(y)$ for an unknown function $f \\colon (0, \\infty) \\to \\R$.\nIt asks which functions turn products into products, that is, which functions are multiplicative.\nIts continuous solutions are the zero function and the power functions $f(x) = x^c$ with $c \\in \\R$.\nIt is the fourth of Cauchy's basic equations, beside the additive, exponential and logarithmic ones.\n\n## Explanations\n\n### L1 human\n\nScaling the input by a factor scales the output by a matching factor.\nThe area of a square of side $x$ is $x^2$, and doubling the side multiplies the area by $2^2 = 4$.\nThe function $f(x) = x^2$ thus satisfies $f(xy) = f(x) f(y)$.\nThis equation is the rule behind scaling laws in geometry and physics.\n",
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