Logarithms: Inverse of Exponential

Description

This animation establishes the relationship between exponential and logarithmic functions. It plots y = 2^x and y = log₂(x) side-by-side reflected across y = x, demonstrating inverse symmetry. A concrete numeric example (2³ = 8 → log₂(8) = 3) is shown with perpendicular guide lines, making the inverse relationship tangible.

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Phases

#	Phase Name	Duration	Description
1	Setup	4s	Draw axes, y = x dashed line
2	Exponential curve	6s	Plot y = 2^x; label it; show a few key points
3	Log curve	6s	Plot y = log₂(x) as reflection; show reflection lines
4	Numeric example	10s	Trace 2³=8: draw vertical at x=3 up to exp curve, horizontal to y-axis showing y=8; then mirror for log
5	Key properties	8s	Show log properties: log(ab), log(a^n), change of base
6	Summary	6s	Final view with both curves, identity line, key formulas

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Layout

+-----------------------------------------------+
|  y                                             |
|  |         y = 2^x (orange)         /         |
|  |       y = log₂(x) (green)   /              |
|  |    y = x (dashed white)  /                 |
|  |                      /  /                  |
|  +---------- x --------                       |
|                                                |
|  2³ = 8  ↔  log₂(8) = 3  [example callout]   |
+-----------------------------------------------+

Area Descriptions

Area	Content	Notes
Center	Axes + three curves	Symmetric reflection visible
On graph	Key point dots (2³=8, log₂8=3)	Connected by perpendicular lines
Bottom	Property formulas	log(ab) = log(a)+log(b), etc.

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Assets & Dependencies

• Fonts: LaTeX
• Colors: EXP_COLOR = ORANGE, LOG_COLOR = GREEN, IDENTITY_COLOR = WHITE (dashed), BG = #1a1a2e
• Manim version: ManimCE 0.19.1

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Notes

• Axes range x: [-1, 5], y: [-2, 8]
• Use numpy for safe log computation (np.log2); restrict log curve to x > 0
• Reflection lines drawn with DashedLine