Symmetry of Logarithmic Functions

This page discusses the symmetry and transformations of funkcja logarytmiczna (logarithmic functions). It explains how to modify the basic logarithmic function to achieve different symmetrical properties.

The basic form of a logarithmic function is presented as f(x) = log₁ x. This serves as the foundation for understanding various transformations.

Definition: A funkcja logarytmiczna is a function of the form f(x) = logₐ x, where a is the base of the logarithm and x is the input variable.

To reflect the logarithmic function over the x-axis, a negative sign is placed before the entire function. For example, f(x) = -log₂ x is the reflection of f(x) = log₂ x over the x-axis.

Example: The function f(x) = log₂ x becomes f(x) = -log₂ x when reflected over the x-axis.

For symmetry with respect to the y-axis, a negative sign is placed before the x variable inside the logarithm. This transformation is represented as f(x) = log₂(-x).

Highlight: The dziedzina funkcji logarytmicznej (domain of the logarithmic function) changes when applying the transformation f(x) = log₂(-x), as the input must be negative for the function to be defined.

The page also illustrates these transformations graphically, showing how the original function's shape changes with each modification.

Vocabulary: Przesunięcie funkcji logarytmicznej refers to the shifting of the logarithmic function graph along the coordinate axes.

These transformations are crucial in understanding the behavior of logarithmic functions and their applications in various mathematical and real-world scenarios.