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  "body": "\n## Summary\n\nCauchy's exponential equation is the [[functional equation|SRAN4EKWS0]] $f(x + y) = f(x) f(y)$ for an unknown function $f \\colon \\R \\to \\R$.\nIt asks which functions turn sums into products.\nIts continuous solutions are the zero function and the [[exponential functions|0J0FCWV32D]] $x \\mapsto a^x$ with $a \u003e 0$.\nIt is one of the four basic equations of Cauchy, together with the additive, logarithmic and power equations.\n\n## Explanations\n\n### L1 human\n\nGrowth by a fixed factor in equal steps is the picture behind this equation.\nIf a quantity is multiplied by $f(x)$ over a time $x$ and by $f(y)$ over a further time $y$, it is multiplied by $f(x) f(y)$ over the time $x + y$.\nFor example, $f(x) = 2^x$ satisfies $2^{x + y} = 2^x \\cdot 2^y$.\nThe equation says that this rule alone almost determines the function.\n",
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