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Logarithmic Action! Dynamic Illustration of Inverse Properties

Logarithmic Action! Dynamic Illustration of Inverse Properties
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2017-04-27 10:02
FunctionCoordinate GeometryTransformationSlider AnimationGraphing
In collections Exponential & Logarithmic Functions
This interactive GeoGebra example dynamically illustrates two inverse properties of logarithms: log_a(a^x)=x and a^(log_a x)=x. It visually demonstrates the fundamental relationship between exponential and logarithmic functions as inverses, using the line y=x as the symmetry axis. The construction employs text fields and input boxes for user-defined base a and variable x, along with commands such as Intersect, Dilate, Segment, Vector, Translate, and Rotate to build the graphs and show corresponding points. Suitable for high school and college students learning logarithms, this tool is ideal for classroom demonstrations or self-study, reinforcing the concept of inverse functions through dynamic manipulation and visual feedback.

GeoGebra Commands & Parameters

  1. Textfield(a) → InputBox1
  2. Textfield(f_1) → InputBox2
  3. Textfield(f) → InputBox3
  4. Intersect(f, h, 1) → E
  5. Dilate(F, If[0 < d ≤ 1, 1 - d, 0], A) → F'
  6. Segment(F, F') → i
  7. Dilate(A, If[1 < d ≤ 2, 2 - d, 0], B) → A'
  8. Segment(A, A') → j
  9. Dilate(B, If[2 < d ≤ 3, 3 - d, 0], G) → B'
  10. Vector(B, B') → u
  11. Translate(A, u) → A'_1
  12. Translate(A', u) → A''
  13. Segment(A'_1, A'') → j'
  14. Rotate(A'_1, If[3 < d ≤ 4, ((-(d - 3)) * π / 2), ((-π)) / 2], F) → H
  15. Rotate(A'', If[3 < d ≤ 4, ((-(d - 3)) * π / 2), ((-π)) / 2], F) → A'''
  16. Segment(H, A''') → j''
  17. Rotate(A'_1, If[3 < d ≤ 4, ((d - 3) * π / 2), π / 2], F) → I
  18. Rotate(A'', If[3 < d ≤ 4, ((d - 3) * π / 2), π / 2], F) → A'''_1
  19. Dilate(D, If[4 < d ≤ 5, 5 - d, 0], E) → D'
  20. Segment(D, D') → l
  21. Dilate(A, If[1 < d ≤ 2, 2 - d, 0], C) → A'_2
  22. Segment(A, A'_2) → m
  23. Dilate(C, If[2 < d ≤ 3, 3 - d, 0], J) → C'
  24. Vector(C, C') → v
  25. Translate(A, v) → A'_3
  26. Translate(A'_2, v) → K
  27. Segment(A'_3, K) → m'
  28. Rotate(A'_3, If[3 < d ≤ 4, ((-(d - 3)) * π / 2), ((-π)) / 2], F) → L
  29. Rotate(K, If[3 < d ≤ 4, ((-(d - 3)) * π / 2), ((-π)) / 2], F) → K'
  30. Segment(L, K') → m''
  31. Intersect(f, p, 1) → N
  32. Dilate(M, If[4 < d ≤ 5, 5 - d, 0], Q) → M'
  33. Segment(M, M') → q
  34. Rotate(F, If[5 < d ≤ 6, ((d - 5) * π / 2), π / 2], F) → F'_1
  35. Rotate(F', If[5 < d ≤ 6, ((d - 5) * π / 2), π / 2], F) → F''
  36. Segment(F'_1, F'') → i'
  37. Dilate(F, If[6 < d ≤ 7, 7 - d, 0], M) → F'_2
  38. Vector(F, F'_2) → w
  39. Translate(F'_1, w) → P
  40. Translate(F'', w) → F'''
  41. Segment(P, F''') → i''