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  "body": "\n## Summary\n\nCauchy's logarithmic 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 sums.\nIts continuous solutions are the functions $f(x) = c \\ln x$ with $c \\in \\R$, the multiples of the [[natural logarithm|04DXAFN2ES]].\nIt is the equation behind slide rules, decibels and other logarithmic scales.\n\n## Explanations\n\n### L1 human\n\nA logarithm turns multiplication into addition.\nThis equation takes that rule as the starting point and asks which functions obey it.\nFor example, $f(x) = \\log_{10} x$ gives $\\log_{10}(100 \\cdot 1000) = 5 = 2 + 3$.\nUp to a constant factor, the logarithm is the only reasonable function with this property.\n",
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