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  "body": "\n## Summary\n\nA logarithmically convex function is a positive function whose logarithm is a [[convex function|07Q7CNM5WF]].\nEvery such function is itself convex, but the converse fails.\nThe best known example is the [[gamma function|0H5SYBSED9]] on the positive reals, and log-convexity is the key hypothesis of the [[Bohr–Mollerup theorem|QFF87AXWN2]].\nThe [[definition|E97P413DKM]] gives the precise condition.\n\n## Explanations\n\n### L1 human\n\nTake a positive function and look at the graph of its logarithm.\nIf that graph bends upward everywhere, the function is log-convex.\nThis is a strong form of convexity: between any two points, the graph lies below the exponential curve through the two end points.\nFor example $e^{x^2}$ is log-convex, since its logarithm is $x^2$.\nThe function $x^2$ on $(0, \\infty)$ is convex but not log-convex, since its logarithm $2 \\ln x$ bends downward.\n",
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