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  "body": "\n## Example\n\nLet $(\\mathbb{R}, +)$ be the [[real numbers|08J2M1JYAG]] with addition, and let $(\\mathbb{R}_{\u003e0}, \\times)$ be the positive real numbers with multiplication.\nThe [[exponential function|0RMVS7RXZX]] gives a map $\\exp : \\mathbb{R} \\to \\mathbb{R}_{\u003e0}$ with $\\exp(x + y) = \\exp(x)\\exp(y)$. Thus it is a homomorphism.\nIts kernel is $\\{0\\}$, because $\\exp(x) = 1$ only for $x = 0$. Thus the map is [[injective|0N2P0XRTDF]].\nEach $y \u003e 0$ is $\\exp(\\ln y)$, so the map is onto. Thus $\\exp$ is an [[isomorphism|085PND4KNT]].\n\n## Remarks\n\nThe inverse isomorphism is the natural logarithm.\n",
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