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  "body": "\n## Proof\n\nWe write $x$ as the product of $x/y$ and $y$.\n\n1. Let $x \u003e 0$ and $y \u003e 0$, and let $\\ln$ be the [[natural logarithm|0044CPHJBV]]. Then $\\frac{x}{y} \u003e 0$, so $\\ln \\frac{x}{y}$ is defined.\n2. We have $x = \\frac{x}{y} \\cdot y$, a product of two positive numbers.\n3. By the [[logarithm of a product|0K2FNY6DYV]], $\\ln x = \\ln \\frac{x}{y} + \\ln y$.\n4. Subtract $\\ln y$ from both sides. This gives $\\ln \\frac{x}{y} = \\ln x - \\ln y$.\n\n## Idea\n\nDivision by $y$ undoes multiplication by $y$. Since $\\ln$ turns products into sums, it turns quotients into differences.\n",
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