Math 2.0

statement proposition

The logarithm is strictly concave

Statement

The natural logarithm is strictly concave on (0,∞)(0, \infty). That is, ln⁡(tx+(1−t)y)>tln⁡x+(1−t)ln⁡y\ln(t x + (1 - t) y) > t \ln x + (1 - t) \ln y for all x≠yx \ne y in (0,∞)(0, \infty) and t∈(0,1)t \in (0, 1).

Remarks

The derivative of ln⁡x\ln x is 1/x1/x, and its second derivative is −1/x2<0-1/x^2 < 0. The concavity gives the AM–GM inequality through Jensen's inequality. Mathlib states it as strictConcaveOn_log_Ioi.

Explanations

L1 human

The graph of ln⁡x\ln x bends downward everywhere. Every chord of the graph lies below the graph.

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What this needs

To read this node, read these 61 nodes in this order. The foundations come first, and the list ends with this node.

  1. Convex function definition
  2. Limit point of a set definition
  3. Local maximum definition
  4. Sequence definition
  5. Strictly monotone function definition
  6. Concave function definition
  7. Limit of a function definition
  8. Local minimum definition
  9. Real sequence definition
  10. Subsequence definition
  11. Continuity at a point definition
  12. Derivative definition
  13. One-sided limit definition
  14. Bounded sequence definition
  15. Geometric sequence definition
  16. Limit of a sequence definition
  17. Monotone sequence definition
  18. Series definition
  19. Continuous function definition
  20. Epsilon-delta definition of continuity statement
  21. Fermat's theorem on stationary points statement
  22. Sum of a finite geometric series statement
  23. Monotone convergence theorem for sequences statement
  24. Convergent sequence definition
  25. Every sequence has a monotone subsequence statement
  26. Squeeze theorem statement
  27. Cauchy product definition
  28. Partial sum definition
  29. Power series definition
  30. Intermediate value theorem statement
  31. Composition of continuous functions is continuous statement
  32. One-sided limits of a monotone function exist statement
  33. Bolzano–Weierstrass theorem statement
  34. The sequence r^n tends to zero for |r| < 1 example
  35. Radius of convergence definition
  36. Convergent series definition
  37. Chain rule statement
  38. Boundedness theorem statement
  39. Continuity of the inverse of a strictly monotone continuous function statement
  40. A series of nonnegative terms converges iff its partial sums are bounded statement
  41. Absolute convergence definition
  42. Sum of a series definition
  43. Extreme value theorem statement
  44. Derivative of an inverse function statement
  45. Comparison test statement
  46. Geometric series example
  47. Cauchy product of absolutely convergent series statement
  48. Rolle's theorem statement
  49. Ratio test statement
  50. Mean value theorem statement
  51. Ratio formula for the radius of convergence statement
  52. A function with positive derivative is strictly increasing statement
  53. The exponential series has infinite radius of convergence statement
  54. Exponential function definition
  55. Exponential of a sum statement
  56. The exponential function is positive statement
  57. The exponential function is its own derivative statement
  58. The exponential function is strictly increasing statement
  59. Natural logarithm definition
  60. Derivative of the logarithm statement
  61. The logarithm is strictly concave statement

Sources

  1. Web https://en.wikipedia.org/wiki/Natural_logarithm, In calculus

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