Page 3: Solving Exponential Equations

This page focuses on techniques for solving exponential equations, which are crucial for understanding potenze con esponente reale esercizi svolti (solved exercises with real exponents).

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

The page outlines two main strategies for solving exponential equations:

1. For equations of the form a^f(x) = a^g(x), set f(x) = g(x).
2. For equations of the form a^x = b^x, transform both sides to have the same base.

Example: To solve 3^(x+2) = 9^(x-1), rewrite as 3^(x+2) = (3^2)^(x-1), then equate exponents: x + 2 = 2(x-1).

The page provides several worked examples, demonstrating how to apply these techniques to solve complex exponential equations.

Page 4: Advanced Exponential Equations

This page continues with more complex exponential equations, building on the techniques introduced earlier.

Highlight: When solving exponential equations, it's often helpful to use logarithms or to make substitutions to simplify the problem.

The page presents several challenging examples, including:

• 2^(x^2 - 5x) = 2^x
• 3^(2x+1) + 3^x = 40

For each example, the page provides a step-by-step solution, demonstrating various problem-solving strategies such as:

• Equating exponents
• Using the change of variable technique
• Applying quadratic equation solving methods

Example: To solve 3^(2x+1) + 3^x = 40, let y = 3^x. This transforms the equation to 3y + y = 40, which can be solved as a quadratic equation in y.

These examples help students develop problem-solving skills for tackling complex potenze con esponente reale Zanichelli exercises.

Page 6: Exponential Growth

This page explores the concept of exponential growth, a key application of exponential functions in real-world scenarios.

Definition: Exponential growth occurs when the rate of change of a quantity is proportional to its current value.

The page uses the example of bacterial growth to illustrate exponential growth:

• Starting with 10 bacteria
• Population increases by 120% each hour

Example: The bacterial population after t hours can be modeled by the function m(t) = 10(2.2)^t.

The page demonstrates how to use this model to predict the bacterial population after 8 hours, resulting in approximately 5,488 bacteria.

Highlight: Exponential growth models are widely used in various fields, including biology, finance, and physics, to make predictions and analyze rapidly changing phenomena.

Page 7: Complex Exponential Functions

This final page examines more complex exponential functions, particularly those of the form y = [f(x)]^g(x).

Vocabulary: The dominio funzioni esponenziali (domain of exponential functions) becomes more complex when dealing with composite functions.

The page analyzes three main cases:

1. y = a^f(x), where a > 0
2. y = [f(x)]^a, where a is a real number
3. y = [f(x)]^g(x)

For each case, the page provides examples and explains how to determine the domain of the function.

Example: For y = (x^2 - 5)^√(x^2 - 1), the domain is determined by two conditions:

1. x^2 - 1 ≥ 0 (for the square root in the exponent)
2. x^2 - 5 > 0 (for the base to be positive)

The page concludes by emphasizing the importance of considering both the base and exponent when determining the domain of complex exponential functions.