Math 2.0

statement theorem

The logarithm grows slower than any power

Statement

For every real a>0a > 0, lim⁡x→∞ln⁡xxa=0.\lim_{x \to \infty} \frac{\ln x}{x^a} = 0. That is, ln⁡x=o(xa)\ln x = o(x^a) as x→∞x \to \infty.

Remarks

Substitute x=et/ax = e^{t/a}. Then ln⁡xxa=t/aet\frac{\ln x}{x^a} = \frac{t / a}{e^{t}}, which tends to 00 by exponential growth. In little o notation, ln⁡x=o(x)\ln x = o(x). Mathlib states this case as Real.isLittleO_log_id_atTop. Similarly, xaln⁡x→0x^a \ln x \to 0 as x→0+x \to 0^+.

Explanations

L1 human

The logarithm goes to infinity, but very slowly. Each positive power of xx, even x0.001x^{0.001}, eventually beats it.

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What this needs

To read this node, read these 55 nodes in this order. The foundations come first, and the list ends with this node.

  1. Big O notation definition
  2. Limit point of a set definition
  3. Local maximum definition
  4. Sequence definition
  5. Strictly monotone function definition
  6. Little o notation 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. Limit at infinity 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. Fermat's theorem on stationary points statement
  21. Sum of a finite geometric series statement
  22. Monotone convergence theorem for sequences statement
  23. Convergent sequence definition
  24. Every sequence has a monotone subsequence statement
  25. Squeeze theorem statement
  26. Cauchy product definition
  27. Partial sum definition
  28. Power series definition
  29. Intermediate value theorem statement
  30. Bolzano–Weierstrass theorem statement
  31. The sequence r^n tends to zero for |r| < 1 example
  32. Radius of convergence definition
  33. Convergent series definition
  34. Boundedness theorem statement
  35. A series of nonnegative terms converges iff its partial sums are bounded statement
  36. Absolute convergence definition
  37. Sum of a series definition
  38. Extreme value theorem statement
  39. Comparison test statement
  40. Geometric series example
  41. Cauchy product of absolutely convergent series statement
  42. Rolle's theorem statement
  43. Ratio test statement
  44. Mean value theorem statement
  45. Ratio formula for the radius of convergence statement
  46. A function with positive derivative is strictly increasing statement
  47. The exponential series has infinite radius of convergence statement
  48. Exponential function definition
  49. Exponential growth beats polynomial growth statement
  50. Exponential of a sum statement
  51. The exponential function is positive statement
  52. The exponential function is its own derivative statement
  53. The exponential function is strictly increasing statement
  54. Natural logarithm definition
  55. The logarithm grows slower than any power statement

Sources

  1. Web https://en.wikipedia.org/wiki/Natural_logarithm, Definition and properties

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