Page 2: Properties of Logarithms

This page delves into the proprietà dei logaritmi, which are crucial for solving logarithmic equations and simplifying expressions.

Key properties include:

1. log_a 1 = 0
2. log_a a = 1
3. a^(log_a b) = b

Vocabulary: The logaritmo definizione matematica states that logarithms and exponentials are inverse functions.

The page introduces three important rules:

1. Logarithm of a Product: log_a (b * c) = log_a b + log_a c
2. Logarithm of a Quotient: log_a (b / c) = log_a b - log_a c
3. Logarithm of a Power: log_a (b^m) = m * log_a b

Example: log_2 (8 * 4) = log_2 8 + log_2 4 = 3 + 2 = 5

The page concludes with the change of base formula, which allows conversion between logarithms of different bases:

log_a b = (log_c b) / (log_c a)

This formula is particularly useful when using calculators, as most support only natural (base e) or common (base 10) logarithms.

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 5: Graphing Logarithmic Functions

This page delves into the graphical representation of logarithmic functions, focusing on how to interpret and sketch grafici funzioni logaritmiche.

Key points:

• The shape of the grafico logaritmo base 10 is similar for all bases > 1, but flipped horizontally for 0 < base < 1.
• The y-intercept is always at (1, 0) for any base.
• For bases > 1, the function increases slowly, while for 0 < base < 1, it decreases slowly.

Example: The grafico logaritmo base 1/2 is a decreasing function that approaches negative infinity as x approaches zero from the right.

The page also covers how to determine if a logarithmic expression is positive or negative based on its graph:

• For bases > 1, values above the x-axis are positive, below are negative.
• For 0 < base < 1, this is reversed.

Highlight: Understanding the grafico esponenziale helps in visualizing logarithmic functions, as they are inverses of each other.

The page concludes with examples of solving simple logarithmic inequalities graphically.

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.

Conclusione

La comprensione approfondita della definizione di logaritmi e proprietà, insieme alla padronanza delle funzioni esponenziali e logaritmiche, è essenziale per affrontare problemi matematici avanzati. Il calcolo del dominio delle funzioni logaritmiche è un passaggio cruciale che richiede attenzione alle condizioni di esistenza. Queste competenze sono fondamentali non solo in matematica pura, ma anche in molte applicazioni pratiche, dalla fisica all'economia, dove i modelli logaritmici ed esponenziali giocano un ruolo chiave.

Page 7: Logarithmic Inequalities

The page introduces Disequazioni logaritmiche schema and techniques for solving logarithmic inequalities.

Definition: Logarithmic inequalities follow similar principles to equations but require attention to sign changes.

Example: Solving log₂(x+2) = 1

Highlight: The importance of considering the base when determining inequality direction.

Page 1: Introduction to Logarithms

The first page introduces the concept of logarithms as a solution to equations where traditional methods fall short. It explains the definizione logaritmo and its basic properties.

Definition: A logarithm is the exponent to which a base must be raised to produce a given number. Mathematically, if a^x = b, then x = log_a b.

Key points:

• Logarithms are defined only for positive arguments (b > 0).
• The base (a) must be positive and not equal to 1.
• If no base is specified, it's assumed to be 10.

Example: log_2 8 = 3 because 2^3 = 8

The page also notes that logarithms can be irrational numbers when the argument is not a rational power of the base.

Highlight: Both the base and the argument of a logarithm must be positive, and the base cannot be 1.