Solving Logarithmic Equations

This page focuses on równania logarytmiczne (logarithmic equations) and demonstrates methods for solving them. It builds upon the concepts introduced in the previous page, applying the properties of logarithms to solve equations.

The page begins with a simple example: log_x 4 = 2. The solution process involves recognizing that this equation is equivalent to x² = 4, leading to the solution x = 2.

Example: Solving log_x 4 = 2

1. Rewrite as x² = 4
2. Solve to get x = 2

Another example presented is log_x 9 = -2. This equation is solved by recognizing that it can be rewritten as x⁻² = 9, leading to the solution x = 3.

Highlight: When solving logarithmic equations, it's often helpful to rewrite the equation in exponential form.

The page emphasizes the importance of checking solutions, as some methods may introduce extraneous solutions that don't satisfy the original equation.

Vocabulary: Extraneous solutions are apparent solutions to an equation that don't actually satisfy the original equation when checked.

While the page doesn't go into extensive detail on more complex logarithmic equations, it provides a solid foundation for understanding the basic principles of solving równania logarytmiczne (logarithmic equations).

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.