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  "body": "\n## Definition\n\n*Cauchy's logarithmic equation* is the [[functional equation|SRAN4EKWS0]]\n$$f(xy) = f(x) + f(y) \\quad \\text{for all } x, y \u003e 0,$$\nfor an unknown [[function|0EPAT51DXQ]] $f \\colon (0, \\infty) \\to \\R$.\nA solution is a function $f$ that satisfies this identity for all positive $x$ and $y$.\n\n## Remarks\n\nExample: $f(x) = \\log_2 x$ is a solution, since $\\log_2(xy) = \\log_2 x + \\log_2 y$.\nNon-example: $f(x) = x - 1$ is not a solution, since $f(2 \\cdot 2) = 3$ while $f(2) + f(2) = 2$.\nThe substitution $x = e^s$, $y = e^t$ turns the equation into additivity of $A(s) = f(e^s)$.\nSo the solutions are $f(x) = A(\\ln x)$ with $A$ an [[additive function|KF719SZD5T]], and the continuous ones are $f(x) = c \\ln x$.\nOn the domain $\\R \\setminus \\set{0}$ the solutions are $f(x) = A(\\ln \\abs{x})$; on all of $\\R$ only $f = 0$ is a solution, since $f(0) = f(0) + f(x)$.\n",
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