Math 2.0

Real functions Theorem

The logarithm grows slower than any power

Builds on Exponential growth beats polynomial growthLittle o notationNatural logarithm

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^+.

Reading path

To understand this page from the ground up, read these 55 pages in order. The foundations come first.

  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

Further reading

  1. Web https://en.wikipedia.org/wiki/Natural_logarithm, Definition and properties
About this page
Tag00DGYY10FV
URIm2://00DGYY10FV
Kindstatement, theorem
WritingW0 stub
MSC26A12, 33B10
StatusThis copy of the site has no build data. Run m2 status 00DGYY10FV.
UsesExponential growth beats polynomial growth, Little o notation, Natural logarithm

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