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  "body": "\n## Statement\n\nLet $\\overline{\\Q}$ be the field of complex [[algebraic numbers|0MNF0BYFQC]].\nLet $\\alpha_1, \\dots, \\alpha_n$ be nonzero algebraic numbers, and let $\\lambda_i$ be a value of the [[complex logarithm|02QGGD5G87]] of $\\alpha_i$ for each $i$.\nSuppose that $\\lambda_1, \\dots, \\lambda_n$ are [[linearly independent|0CRMGPAH83]] over $\\Q$.\nThen $1, \\lambda_1, \\dots, \\lambda_n$ are linearly independent over $\\overline{\\Q}$.\n\n## Remarks\n\nThe proof extends the method of Gelfond.\nIt builds an auxiliary function in several variables that vanishes to high order at many points, then uses an extrapolation step and an estimate to reach a contradiction.\nThe method also gives an effective lower bound.\nLet $\\Lambda = \\beta_0 + \\beta_1 \\lambda_1 + \\dots + \\beta_n \\lambda_n$ with algebraic $\\beta_i$, and suppose $\\Lambda \\ne 0$.\nThen $\\abs{\\Lambda}$ is bounded below by an explicit function of the degrees and heights of the $\\alpha_i$ and $\\beta_i$.\nThe [[Gelfond–Schneider theorem|RN58Z2NWAC]] is a special case.\nAnother consequence: $e^{\\beta_0} \\alpha_1^{\\beta_1} \\cdots \\alpha_n^{\\beta_n}$ is [[transcendental|0VTQXTCR2G]] for nonzero algebraic $\\alpha_i$ and $\\beta_i$.\nFor example, $\\pi + \\log 2$ is transcendental: indeed $\\pi = -i \\log(-1)$, and $\\log(-1) = i\\pi$ and $\\log 2$ are linearly independent over $\\Q$.\nThe hypothesis of independence over $\\Q$ is necessary: $\\log 4 - 2 \\log 2 = 0$ is a relation with algebraic coefficients.\n\n## Explanations\n\n### L1 human\n\nLogarithms of algebraic numbers behave as independently as they possibly can.\nIf no relation with rational coefficients links them, then no relation with algebraic coefficients does either.\nThe quantitative form says that such a combination, when it is not zero, cannot be extremely close to zero.\n\n### L2 human\n\nThe effective lower bounds for linear forms in logarithms are the key input for many effective results in Diophantine equations.\nThey give explicit bounds for the solutions of [[Thue equations|3Q6CFPQZDF]] and of $S$-unit equations.\nThey were also used in the solution of the [[class number problem|KTGAS9DGRX]] for imaginary quadratic fields of class number one.\nThe theorem is a step toward [[Schanuel's conjecture|65PV4VFP9Y]], which would give algebraic independence instead of linear independence.\n",
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