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  "body": "\n## Proof\n\nWe apply the exponential function and use that it turns sums into products.\n\n1. Let $x \u003e 0$ and $y \u003e 0$, and let $\\ln$ be the [[natural logarithm|0044CPHJBV]]. Then $x y \u003e 0$, so $\\ln(x y)$ is defined.\n2. By the [[exponential of a sum|0E8HNVBCSP]], $e^{\\ln x + \\ln y} = e^{\\ln x} \\, e^{\\ln y}$.\n3. By the [[inverse relation|0RKW2X63YR]] $e^{\\ln t} = t$ for $t \u003e 0$, the right side of step 2 equals $x y$.\n4. Apply $\\ln$ to both sides of $e^{\\ln x + \\ln y} = x y$. The inverse relation $\\ln(e^s) = s$ for real $s$ gives $\\ln x + \\ln y = \\ln(x y)$.\n\n## Idea\n\nThe exponential function turns a sum into a product, and $\\ln$ undoes the exponential function. So $\\ln$ turns a product into a sum.\n",
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