Applications of Logarithmic Functions: Geometric Mean & Expected Value

This GeoGebra example demonstrates how logarithmic functions turn multiplicative relationships into additive ones, using a visual construction that connects log models with geometric mean–type quantities and an expected-value style aggregation. The key math concepts include the function form log(x), logarithm properties (log of products/powers leading to sums and multiples), working with fractional or ratio quantities via logarithms, the geometric mean interpretation through log-linear relationships, and an expected value viewpoint expressed as a weighted sum of relevant (log-transformed) quantities. Construction-wise, the setup is organized around the GeoGebra objects listed in the construction summary—points A1–A9, B1, and B2—where the A points serve as geometric/reference positions for the represented quantities and B1/B2 act as additional anchor or comparison objects. Using these objects, the figure supports creating the log-related graphical components (e.g., function/graph elements and measurement-linked segments) that show the before/after correspondence under log transformation. Suitable for middle-to-high school advanced learners and early college students studying functions, helping students build intuition and practice applying logarithmic properties in contexts involving geometric means and expectation-like computations.