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Logarithmic Action (3)! – Dynamic Discovery of the Power Property

Logarithmic Action (3)! – Dynamic Discovery of the Power Property
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2019-06-27 01:22
FunctionCoordinate GeometryTransformationSlider AnimationGraphing
In collections Exploring Exponential & Logarithmic Functions Exploring Exponential & Logarithmic Functions
① This interactive GeoGebra construction visually demonstrates the power property of logarithms (log_b(a^n) = n·log_b(a)) through a dynamic, modifiable discovery environment. ② Core mathematical concepts include logarithmic functions, the power rule for logarithms, function transformations, and coordinate geometry involving points, segments, vectors, and distances. ③ The construction employs GeoGebra commands such as Textfield, InputBox, Intersect, Dilate, Segment, Vector, Translate, Rotate, Distance, Midpoint, Line, and Circle to create a rich set of objects (points A, B, C, D, E, F and their primed variants) that respond to user input, allowing real-time comparison of the two sides of the logarithmic identity. ④ This resource is ideal for high school students learning logarithmic properties, as well as for teachers seeking an interactive demonstration to foster conceptual understanding and inquiry-based exploration.

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. Dilate(A, If[1 < d ≤ 2, 2 - d, 0], B) → A'
  7. Segment(A, A') → j
  8. Dilate(B, If[2 < d ≤ 3, 3 - d, 0], G) → B'
  9. Vector(B, B') → u
  10. Translate(A, u) → A'_1
  11. Translate(A', u) → A''
  12. Segment(A'_1, A'') → j'
  13. Rotate(A'_1, If[3 < d ≤ 4, ((-(d - 3)) * π / 2), ((-π)) / 2], F) → H
  14. Rotate(A'', If[3 < d ≤ 4, ((-(d - 3)) * π / 2), ((-π)) / 2], F) → A'''
  15. Segment(H, A''') → j''
  16. Rotate(A'_1, If[3 < d ≤ 4, ((d - 3) * π / 2), π / 2], F) → I
  17. Rotate(A'', If[3 < d ≤ 4, ((d - 3) * π / 2), π / 2], F) → A'''_1
  18. Dilate(D, If[4 < d ≤ 5, 5 - d, 0], E) → D'
  19. Segment(D, D') → l
  20. Dilate(A, If[1 < d ≤ 2, 2 - d, 0], C) → A'_2
  21. Segment(A, A'_2) → m
  22. Dilate(A, If[2 < d ≤ 3, 3 - d, 0], R) → C'
  23. Vector(C, C') → v
  24. Intersect(f, p, 1) → N
  25. Rotate(F, If[5 < d ≤ 6, ((d - 5) * π / 2), π / 2], F) → F'_1
  26. Rotate(F', If[5 < d ≤ 6, ((d - 5) * π / 2), π / 2], F) → F''
  27. Dilate(F, If[6 < d ≤ 7, 7 - d, 0], M) → F'_2
  28. Vector(F, F'_2) → w
  29. Translate(F'_1, w) → P
  30. Translate(F'', w) → F'''
  31. Distance(A, C) → distanceAC
  32. Vector(A, A'_2) → i
  33. Vector(A, C') → b_1
  34. Midpoint(Z, A_1) → Q
  35. Line(B_1, C_1) → e
  36. Segment(B_1, C_1) → g_1
  37. Circle(B_1, g_1 / 3) → c_1
  38. Intersect(c_1, e, 2) → T
  39. Circle(T, g_1 / 3) → d_1
  40. Intersect(d_1, e, 2) → U
  41. Dilate(Z, If[2 < d ≤ 3, 3 - d, 0], Q) → Z'
  42. Segment(Z, Z') → q
  43. Dilate(Q, If[3 < d ≤ 4, 4 - d, 0], A_1) → Q'
  44. Segment(Q, Q') → h_1
  45. Dilate(B_1, If[4 < d ≤ 5, 5 - d, 0], T) → D_1
  46. Segment(B_1, D_1) → i_1
  47. Dilate(T, If[5 < d ≤ 6, 6 - d, 0], U) → T'
  48. Segment(T, T') → j_1
  49. Dilate(U, If[6 < d ≤ 7, 7 - d, 0], C_1) → U'
  50. Segment(U, U') → k_1
  51. Segment(A, A'_2) → l_1
  52. Distance(Z, A_1) → distanceZA1
  53. Distance(B_1, C_1) → distanceB1C1
  54. Dilate(A, If[7 < d ≤ 8, 8 - d, 0], Z) → A'_3
  55. Vector(A, A'_3) → u_1
  56. Dilate(A, If[7 < d ≤ 8, 8 - d, 0], B_1) → A'_4
  57. Vector(A, A'_4) → v_1
  58. Translate(A, u_1) → A'_5
  59. Translate(A'_2, u_1) → L
  60. Segment(A'_5, L) → m_1
  61. Translate(A, v_1) → A'_6
  62. Translate(A'_2, v_1) → W
  63. Segment(A'_6, W) → n_1
  64. Rotate(A'_5, If[8 < d ≤ 9, ((d - 8) * π), π], Q) → E_1
  65. Rotate(L, If[8 < d ≤ 9, ((d - 8) * π), π], Q) → L'
  66. Segment(E_1, L') → p_1
  67. Rotate(A'_6, If[8 < d ≤ 9, ((d - 8) * π), π], T) → F_1
  68. Rotate(W, If[8 < d ≤ 9, ((d - 8) * π), π], T) → W'
  69. Segment(F_1, W') → q_1
  70. Rotate(F_1, If[9 < d ≤ 10, ((d - 9) * π), π], U) → G_1
  71. Rotate(W', If[9 < d ≤ 10, ((d - 9) * π), π], U) → W''
  72. Segment(G_1, W'') → r_1

AI Drawing Prompt Reference

Create a dynamic illustration showing the power property of logarithms with two adjustable inputs and a logarithmic function graph. Display two geometric constructions involving scaled and rotated points, segments, and vectors that correspond to both sides of the logarithmic identity. The illustration should update in real-time as inputs change, clearly showing the relationship between the two constructions to visually prove the power rule for logarithms.
The prompt above was compiled from the original drawing commands and is for reference only.