Page 3: Logarithmic Functions

This page explores the funzione logaritmica and its properties, comparing it to its inverse, the exponential function.

Key properties of logarithmic functions:

• Domain: All positive real numbers (x > 0)
• Range: All real numbers
• The function is increasing when a > 1 and decreasing when 0 < a < 1
• The graph always passes through the point (1, 0)

Definition: The logaritmo definizione semplice states that y = log_a x is the inverse function of y = a^x.

The page discusses transformations of logarithmic functions:

• Vertical and horizontal translations
• Reflections over the x-axis, y-axis, and origin

Example: y = ln(x - 1) + 2 represents a logarithmic function shifted 1 unit right and 2 units up.

The page concludes with a discussion on finding the domain of logarithmic functions, emphasizing the importance of keeping the argument strictly positive.

Highlight: The grafico logaritmo naturale is a special case where the base is e (≈ 2.71828).

Page 4: Domains of Logarithmic Functions

This page focuses on determining the domains of more complex logarithmic functions, including those with multiple terms and fractions.

Key points:

• For simple logarithmic functions, ensure the argument is strictly positive.
• For functions with multiple logarithmic terms, consider each term separately and then combine the results.
• For logarithmic functions with fractions, ensure both the argument of the logarithm and the denominator are non-zero.

Example: For y = log(x + 5) + log(x - 3), the domain is x > 3 because both (x + 5) and (x - 3) must be positive.

The page also covers more complex scenarios, such as:

• Logarithms of quadratic expressions
• Logarithms with absolute value arguments

Highlight: When dealing with equazioni e disequazioni logaritmiche, always check the domain to ensure the logarithms are defined.

The page concludes with a brief discussion on limits of logarithmic functions as x approaches infinity and zero.

Page 6: Logarithmic Equations

The final page focuses on solving equazioni e disequazioni logaritmiche, providing a systematic approach to these problems.

General form of a logarithmic equation: log_a A(x) = log_b B(x)

Steps to solve:

1. Use logarithm properties to simplify if possible.
2. Apply the change of base formula if needed.
3. If the bases are the same, set the arguments equal: A(x) = B(x).
4. Solve the resulting equation.
5. Check solutions in the original equation to avoid extraneous roots.

Highlight: Always check the domain conditions (A(x) > 0 and B(x) > 0) when solving logarithmic equations.

Example: Solve log_2(x+1) + log_2(x-1) - 3 = 0 Solution:

1. Combine logarithms: log_2((x+1)(x-1)) = 3
2. Simplify: log_2(x^2-1) = 3
3. Apply exponential: 2^3 = x^2-1
4. Solve: x = ±√9 = ±3
5. Check: x = 3 is the only valid solution as x > 1 is required for the domain.

The page emphasizes the importance of understanding proprietà logaritmi and equazioni logaritmiche schema for efficiently solving these types of problems.