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Scopri le Equazioni e Disequazioni Esponenziali e Logaritmiche: Teoria, Esercizi e Mappe!

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Scopri le Equazioni e Disequazioni Esponenziali e Logaritmiche: Teoria, Esercizi e Mappe!

Rita Cornacchini

@ritacornacchini_zcpf

261 Follower

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Exponential and Logarithmic Functions: A Comprehensive Guide

This guide covers the essential concepts of exponential and logarithmic functions, including equations, properties, and graphical representations. It provides a thorough understanding of these mathematical concepts for students.

• Explores exponential equations and their solutions
• Examines logarithmic functions and their properties
• Discusses exponential and logarithmic inequalities
• Includes graphical representations and practical examples

15/9/2022

6590

Eq. esponenziale
(elementare)
a* = b
RIASSUNTO MATE 1
ESPONEN2IALI
a>o
a#1
b>0
Sotto queste
ed è pari a
a
2 a f(x)
In generale le eq. espone

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Logarithmic Equations

This page focuses on logarithmic equations, which are equations where the unknown variable appears as the argument of a logarithm. The page begins by introducing the elementary logarithmic equation:

Definition: An elementary logarithmic equation has the form log_a x = b, with the solution x = a^b, where x > 0.

The page then outlines three general types of logarithmic equations and their solving methods:

log_a f(x) = k
f(x) = log_a g(x)
a_1 log_a x + a_2 log_a x + a_3 = 0

Example: To solve log_2 x = 3, we get x = 2^3 = 8.

The solving process often involves substitution and solving resulting quadratic equations. The page emphasizes the importance of checking solutions against the domain restrictions of logarithms.

Highlight: Always remember the condition x > 0 when solving logarithmic equations, as logarithms are only defined for positive arguments.

Vedi

Exponential Equations and Functions

Exponential equations are a fundamental concept in mathematics, particularly in the study of exponential functions. This page introduces the basic form of exponential equations and their solutions.

An elementary exponential equation takes the form a^x = b, where a > 0, a ≠ 1, and b > 0. Under these conditions, the solution is unique and equal to x = log_a b. The page also outlines three general types of exponential equations and their solving methods.

Definition: An exponential equation is an equation where the variable appears in the exponent.

Example: For the equation 2^x = 8, the solution is x = log_2 8 = 3.

The page concludes with a brief introduction to exponential functions, noting that their graphs depend on the base a.

Highlight: Exponential functions with a base greater than 1 are increasing functions, while those with a base between 0 and 1 are decreasing functions.

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Exponential Functions Graphical Representation

This page focuses on the graphical representation of exponential functions, which take the form y = a^x. The shape of the graph depends on the value of the base a.

For a > 1, the function is increasing, with the graph rising from left to right. For 0 < a < 1, the function is decreasing, with the graph falling from left to right. In both cases, the domain of the function is all real numbers (R).

Highlight: The y-axis (x = 0) is always an asymptote for exponential functions.

Example: The graph of y = 2^x is an increasing exponential function, while y = (1/2)^x is a decreasing exponential function.

These graphical representations help visualize the behavior of exponential functions and their key properties.

Vedi

Logarithm Theorems and Change of Base Formula

This page delves into the important theorems of logarithms and introduces the change of base formula. These properties are crucial for simplifying and solving logarithmic expressions and equations.

Highlight: Key logarithm properties include:

log_a (b·c) = log_a b + log_a c

log_a (b/c) = log_a b - log_a c

log_a (b^m) = m · log_a b

The change of base formula is presented as:

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

This formula allows for the conversion between logarithms of different bases, which is particularly useful when working with calculators or when simplifying complex logarithmic expressions.

Example: log_2 8 can be calculated using the natural logarithm as ln 8 / ln 2.

These properties and the change of base formula are essential tools for manipulating and solving logarithmic functions and equations.

Vedi

Logarithms: Definition and Properties

This page introduces the concept of logarithms and their fundamental properties. Logarithms are defined as the inverse operation of exponentiation.

Definition: The logarithm of a number b with base a, written as log_a b, is the exponent to which a must be raised to produce b.

The page outlines the conditions for logarithms: a > 0, a ≠ 1, and b > 0. It also introduces special logarithms such as the natural logarithm (base e) and the common logarithm (base 10).

Highlight: A logarithm is positive when both the base and the argument are greater than 1 or when both are between 0 and 1.

The page concludes with important relationships between logarithms and exponents, such as a^(log_a b) = b and log_a (a^x) = x.

Vedi

Exponential and Logarithmic Inequalities

This final page covers exponential and logarithmic inequalities, which are crucial in solving many real-world problems involving growth and decay.

For exponential inequalities:

a^(f(x)) > k becomes f(x) > log_a k for a > 1
a^(f(x)) < k becomes f(x) < log_a k for 0 < a < 1

Example: Solve 2^x > 8. This becomes x > log_2 8 = 3.

For logarithmic inequalities:

log_a f(x) > k becomes f(x) > a^k for a > 1
log_a f(x) < k becomes 0 < f(x) < a^k for a > 1

Highlight: Always consider the domain restrictions when solving logarithmic inequalities: f(x) > 0.

The page also includes compound inequalities and systems of inequalities involving exponential and logarithmic functions.

Vocabulary: A compound inequality involves two or more inequalities combined with "and" or "or" statements.

This comprehensive overview of exponential and logarithmic inequalities completes the guide, providing students with the tools to tackle a wide range of problems in this important area of mathematics.

Vedi

Logarithmic Functions

This page explores logarithmic functions, which are the inverse of exponential functions. The general form of a logarithmic function is y = log_a x.

The graph of a logarithmic function depends on the base a:

For a > 1, the function is increasing.
For 0 < a < 1, the function is decreasing.

Highlight: The domain of all logarithmic functions is x > 0, and the y-axis (x = 1) is always an asymptote.

Example: The natural logarithm function y = ln x is an increasing logarithmic function with base e ≈ 2.718.

The page includes graphical representations of logarithmic functions, illustrating their key features such as the y-intercept at (1,0) and their behavior as x approaches 0 and positive infinity.

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Knowunity è l'app per l'istruzione numero 1 in cinque paesi europei

Knowunity è stata inserita in un articolo di Apple ed è costantemente in cima alle classifiche degli app store nella categoria istruzione in Germania, Italia, Polonia, Svizzera e Regno Unito. Unisciti a Knowunity oggi stesso e aiuta milioni di studenti in tutto il mondo.

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App Store

Knowunity è l'app per l'istruzione numero 1 in cinque paesi europei

4.9+

Valutazione media dell'app

13 M

Studenti che usano Knowunity

#1

Nelle classifiche delle app per l'istruzione in 12 Paesi

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Studenti che hanno caricato appunti

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Utente iOS

Adoro questa applicazione [...] consiglio Knowunity a tutti!!! Sono passato da un 5 a una 8 con questa app

Stefano S, utente iOS

L'applicazione è molto semplice e ben progettata. Finora ho sempre trovato quello che stavo cercando

Susanna, utente iOS

Adoro questa app ❤️, la uso praticamente sempre quando studio.

Materie

Matematica

Relazioni e funzioni

Esponenziali e logaritmi

Scopri le Equazioni e Disequazioni Esponenziali e Logaritmiche: Teoria, Esercizi e Mappe!

Rita Cornacchini

@ritacornacchini_zcpf

261 Follower

Segui

Exponential and Logarithmic Functions: A Comprehensive Guide

15/9/2022

6590

2ªl/3ªl

Matematica

205

Logarithmic Equations

Definition: An elementary logarithmic equation has the form log_a x = b, with the solution x = a^b, where x > 0.

The page then outlines three general types of logarithmic equations and their solving methods:

log_a f(x) = k
f(x) = log_a g(x)
a_1 log_a x + a_2 log_a x + a_3 = 0

Example: To solve log_2 x = 3, we get x = 2^3 = 8.

The solving process often involves substitution and solving resulting quadratic equations. The page emphasizes the importance of checking solutions against the domain restrictions of logarithms.

Highlight: Always remember the condition x > 0 when solving logarithmic equations, as logarithms are only defined for positive arguments.

Exponential Equations and Functions

Definition: An exponential equation is an equation where the variable appears in the exponent.

Example: For the equation 2^x = 8, the solution is x = log_2 8 = 3.

The page concludes with a brief introduction to exponential functions, noting that their graphs depend on the base a.

Highlight: Exponential functions with a base greater than 1 are increasing functions, while those with a base between 0 and 1 are decreasing functions.

Exponential Functions Graphical Representation

This page focuses on the graphical representation of exponential functions, which take the form y = a^x. The shape of the graph depends on the value of the base a.

Highlight: The y-axis (x = 0) is always an asymptote for exponential functions.

Example: The graph of y = 2^x is an increasing exponential function, while y = (1/2)^x is a decreasing exponential function.

These graphical representations help visualize the behavior of exponential functions and their key properties.

Logarithm Theorems and Change of Base Formula

This page delves into the important theorems of logarithms and introduces the change of base formula. These properties are crucial for simplifying and solving logarithmic expressions and equations.

Highlight: Key logarithm properties include:

log_a (b·c) = log_a b + log_a c

log_a (b/c) = log_a b - log_a c

log_a (b^m) = m · log_a b

The change of base formula is presented as:

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

This formula allows for the conversion between logarithms of different bases, which is particularly useful when working with calculators or when simplifying complex logarithmic expressions.

Example: log_2 8 can be calculated using the natural logarithm as ln 8 / ln 2.

These properties and the change of base formula are essential tools for manipulating and solving logarithmic functions and equations.

Logarithms: Definition and Properties

This page introduces the concept of logarithms and their fundamental properties. Logarithms are defined as the inverse operation of exponentiation.

Definition: The logarithm of a number b with base a, written as log_a b, is the exponent to which a must be raised to produce b.

The page outlines the conditions for logarithms: a > 0, a ≠ 1, and b > 0. It also introduces special logarithms such as the natural logarithm (base e) and the common logarithm (base 10).

Highlight: A logarithm is positive when both the base and the argument are greater than 1 or when both are between 0 and 1.

The page concludes with important relationships between logarithms and exponents, such as a^(log_a b) = b and log_a (a^x) = x.

Exponential and Logarithmic Inequalities

This final page covers exponential and logarithmic inequalities, which are crucial in solving many real-world problems involving growth and decay.

For exponential inequalities:

a^(f(x)) > k becomes f(x) > log_a k for a > 1
a^(f(x)) < k becomes f(x) < log_a k for 0 < a < 1

Example: Solve 2^x > 8. This becomes x > log_2 8 = 3.

For logarithmic inequalities:

log_a f(x) > k becomes f(x) > a^k for a > 1
log_a f(x) < k becomes 0 < f(x) < a^k for a > 1

Highlight: Always consider the domain restrictions when solving logarithmic inequalities: f(x) > 0.

The page also includes compound inequalities and systems of inequalities involving exponential and logarithmic functions.

Vocabulary: A compound inequality involves two or more inequalities combined with "and" or "or" statements.

Logarithmic Functions

This page explores logarithmic functions, which are the inverse of exponential functions. The general form of a logarithmic function is y = log_a x.

The graph of a logarithmic function depends on the base a:

For a > 1, the function is increasing.
For 0 < a < 1, the function is decreasing.

Highlight: The domain of all logarithmic functions is x > 0, and the y-axis (x = 1) is always an asymptote.

Example: The natural logarithm function y = ln x is an increasing logarithmic function with base e ≈ 2.718.

The page includes graphical representations of logarithmic functions, illustrating their key features such as the y-intercept at (1,0) and their behavior as x approaches 0 and positive infinity.

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Knowunity è l'app per l'istruzione numero 1 in cinque paesi europei

Knowunity è stata inserita in un articolo di Apple ed è costantemente in cima alle classifiche degli app store nella categoria istruzione in Germania, Italia, Polonia, Svizzera e Regno Unito. Unisciti a Knowunity oggi stesso e aiuta milioni di studenti in tutto il mondo.

Scarica

Google Play

Scarica

App Store

4.9+

Valutazione media dell'app

13 M

Studenti che usano Knowunity

#1

Nelle classifiche delle app per l'istruzione in 12 Paesi

950 K+

Studenti che hanno caricato appunti

Non siete ancora sicuri? Guarda cosa dicono gli altri studenti...

Utente iOS

Adoro questa applicazione [...] consiglio Knowunity a tutti!!! Sono passato da un 5 a una 8 con questa app

Stefano S, utente iOS

L'applicazione è molto semplice e ben progettata. Finora ho sempre trovato quello che stavo cercando

Susanna, utente iOS