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  "body": "\n## Definition\n\nLet $z$ be a complex number with $z \\neq 0$.\nA complex logarithm of $z$ is a complex number $w$ with $e^w = z$, where $e^w$ is the [[complex exponential|054VXHMCA3]] of $w$.\nThe notation $\\log z$ means the set of all complex logarithms of $z$, or one chosen element of this set.\n\n## Remarks\n\nLet $|z|$ be the [[modulus|0KW2X0TFBX]] of $z$, let $\\ln$ be the real natural logarithm, and let $\\theta$ be an [[argument|0ZZRSX91QR]] of $z$. Then the logarithms of $z$ are exactly the numbers $\\ln |z| + i (\\theta + 2 \\pi k)$ with $k \\in \\mathbb{Z}$.\nThe number $0$ has no logarithm, because $e^w \\neq 0$ for each $w$.\nSo $\\log$ is a [[multivalued function|03CPD1HGF3]] on $\\mathbb{C} \\setminus \\{0\\}$. Its [[principal branch|0YKY4M3KRQ]] is $\\operatorname{Log} z = \\ln |z| + i \\operatorname{Arg} z$, with $-\\pi \u003c \\operatorname{Arg} z \\le \\pi$.\nFor example, the logarithms of $-1$ are $i \\pi (2k + 1)$ with $k \\in \\mathbb{Z}$, and $\\operatorname{Log}(-1) = i \\pi$.\n\n## Explanations\n\n### L1 human\n\nA logarithm of $z$ is an answer to the question: $e$ to which power gives $z$?\nBecause $e^{w + 2\\pi i} = e^w$, each answer gives infinitely many other answers, spaced by $2\\pi i$ on a vertical line.\nThe real part of each answer is $\\ln |z|$, and the imaginary part is an angle of $z$.\n",
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