When the Exponent Becomes the Output: A Log Function Generator

This example demonstrates a construction where the exponent is treated as the output, allowing students to generate and explore log-related function graphs interactively. The key ideas are the inverse relationship between exponential and logarithmic functions, representing functional dependencies in the coordinate plane, and understanding how parameter changes translate into geometric effects such as dilation and translation. The GeoGebra build leverages commands and objects from the construction summary: Point and Intersect to place critical points and determine intersections; Vector and Segment for directional/auxiliary geometry; RotateText to keep labels readable after transformations; Dilate and Translate as the main tools to map a base configuration into the target one; Circle for reference geometry; and Rotate for additional orientation adjustments. Together with the plotted/generated objects (e.g., g, h, f, p and related constructed points/lines) and interactive controls (buttons and text elements), the user can visually connect the “exponent-as-output” setup to the resulting logarithmic structure. Suitable for high school through early college, especially for lessons on inverse functions, exponential/log transformations, and graphing/log function intuition.