Logarithmic Functions and Their Transformations
This page introduces the concept of funkcja logarytmiczna (logarithmic function) and explores its properties and graphical representations. It covers the basic definition, domain restrictions, and various transformations of the logarithmic function.
The fundamental logarithmic function is defined as f(x) = log₂ x, where the base is 2. The dziedzina funkcji logarytmicznej (domain of the logarithmic function) is restricted to positive real numbers, as logarithms are undefined for non-positive values.
Definition: log₂ b = c means that 2^c = b, where a > 0, a ≠ 1, and b > 0.
The page then delves into przekształcenia wykresów funkcji logarytmicznej (transformations of logarithmic function graphs). It demonstrates how to shift the graph of f(x) = log₂ x vertically and horizontally.
Example: g(x) = log₂ (x+4) represents a horizontal shift of the basic logarithmic function 4 units to the left.
Example: h(x) = log₂ (x+4) - 3 combines a horizontal shift of 4 units left with a vertical shift of 3 units down.
The document also touches on more complex logarithmic functions, such as f(x) = log₃₋ₓ² (2x), highlighting the importance of considering domain restrictions in such cases.
Highlight: For any logarithmic function, it's crucial to remember that log_a 1 = 0 for any base a > 0, a ≠ 1.
The page concludes with a general form for transformed logarithmic functions: f(x) = log_a (x-p) + q, where p represents horizontal shift and q represents vertical shift.
