Math 2.0

definition

Real exponentiation

Definition

Let a>0a > 0 and let xx be a real number. The power axa^x is ax=exln⁡a.a^x = e^{x \ln a}.

Remarks

For natural numbers x=nx = n, this agrees with the product of nn factors aa. For x=p/qx = p/q, it agrees with apq\sqrt[q]{a^p}. The usual rules hold: ax+y=axaya^{x + y} = a^x a^y, (ax)y=axy(a^x)^y = a^{x y}, and (ab)x=axbx(a b)^x = a^x b^x. For a=ea = e, it gives ex=exp⁡(x)e^x = \exp(x), since ln⁡e=1\ln e = 1. Mathlib defines Real.rpow with extra conventions for a≤0a \le 0.

Explanations

L1 human

To raise aa to a real power xx, multiply the logarithm of aa by xx and apply the exponential function. This fills in all powers between the rational ones continuously.

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To read this node, read these 51 nodes in this order. The foundations come first, and the list ends with this node.

  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. Bounded sequence definition
  12. Geometric sequence definition
  13. Limit of a sequence definition
  14. Monotone sequence definition
  15. Series definition
  16. Continuous function definition
  17. Fermat's theorem on stationary points statement
  18. Sum of a finite geometric series statement
  19. Monotone convergence theorem for sequences statement
  20. Convergent sequence definition
  21. Every sequence has a monotone subsequence statement
  22. Squeeze theorem statement
  23. Cauchy product definition
  24. Partial sum definition
  25. Power series definition
  26. Intermediate value theorem statement
  27. Bolzano–Weierstrass theorem statement
  28. The sequence r^n tends to zero for |r| < 1 example
  29. Radius of convergence definition
  30. Convergent series definition
  31. Boundedness theorem statement
  32. A series of nonnegative terms converges iff its partial sums are bounded statement
  33. Absolute convergence definition
  34. Sum of a series definition
  35. Extreme value theorem statement
  36. Comparison test statement
  37. Geometric series example
  38. Cauchy product of absolutely convergent series statement
  39. Rolle's theorem statement
  40. Ratio test statement
  41. Mean value theorem statement
  42. Ratio formula for the radius of convergence statement
  43. A function with positive derivative is strictly increasing statement
  44. The exponential series has infinite radius of convergence statement
  45. Exponential function definition
  46. Exponential of a sum statement
  47. The exponential function is positive statement
  48. The exponential function is its own derivative statement
  49. The exponential function is strictly increasing statement
  50. Natural logarithm definition
  51. Real exponentiation definition

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Sources

  1. Web https://en.wikipedia.org/wiki/Exponentiation, Powers via logarithms

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