Math 2.0

Special functions Theorem

Logarithm is the inverse of the exponential

Builds on Natural logarithmThe exponential function is strictly increasingContinuity of the inverse of a strictly monotone continuous function

Statement

For every real x>0x > 0, eln⁡x=xe^{\ln x} = x. For every real yy, ln⁡(ey)=y\ln(e^y) = y. The function ln⁡\ln is continuous and strictly increasing on (0,∞)(0, \infty), and its range is ℝ\mathbb{R}.

Remarks

The two identities say that ln⁡\ln and exp⁡\exp are inverse bijections between (0,∞)(0, \infty) and ℝ\mathbb{R}. Continuity follows from the continuity of inverse functions, since exp⁡\exp is strictly increasing and continuous. Also ln⁡x→∞\ln x \to \infty as x→∞x \to \infty, and ln⁡x→−∞\ln x \to -\infty as x→0+x \to 0^+. Mathlib states the first identity as Real.exp_log.

Reading path

To understand this page from the ground up, read these 54 pages in order. The foundations come first.

  1. Limit point of a set definition
  2. Local maximum definition
  3. Sequence definition
  4. Strictly monotone function definition
  5. Limit of a function definition
  6. Local minimum definition
  7. Real sequence definition
  8. Subsequence definition
  9. Continuity at a point definition
  10. Derivative definition
  11. One-sided limit definition
  12. Bounded sequence definition
  13. Geometric sequence definition
  14. Limit of a sequence definition
  15. Monotone sequence definition
  16. Series definition
  17. Continuous function definition
  18. Fermat's theorem on stationary points statement
  19. Sum of a finite geometric series statement
  20. Every sequence has a monotone subsequence statement
  21. Monotone convergence theorem for sequences statement
  22. Convergent sequence definition
  23. Squeeze theorem statement
  24. Cauchy product definition
  25. Partial sum definition
  26. Power series definition
  27. Intermediate value theorem statement
  28. One-sided limits of a monotone function exist statement
  29. Bolzano–Weierstrass theorem statement
  30. The sequence r^n tends to zero for |r| < 1 example
  31. Radius of convergence definition
  32. Convergent series definition
  33. Boundedness theorem statement
  34. Continuity of the inverse of a strictly monotone continuous function 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 of a sum statement
  50. The exponential function is positive statement
  51. The exponential function is its own derivative statement
  52. The exponential function is strictly increasing statement
  53. Natural logarithm definition
  54. Logarithm is the inverse of the exponential statement

Where this leads

4 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Natural_logarithm, Definition and properties
About this page
Tag0RKW2X63YR
URIm2://0RKW2X63YR
Kindstatement, theorem
WritingW0 stub
MSC33B10
StatusThis copy of the site has no build data. Run m2 status 0RKW2X63YR.
UsesNatural logarithm, The exponential function is strictly increasing, Continuity of the inverse of a strictly monotone continuous function

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