Lesson Plan | Traditional Methodology | Logarithmic Function: Graph

Objectives

Duration: (10 - 15 minutes)

The purpose of this stage is to prepare students for a detailed and practical understanding of logarithmic functions, focusing on the ability to identify, construct, and interpret their graphs. This foundation is essential for students to advance to solving more complex problems and practical applications of logarithmic functions in mathematical and scientific contexts.

Main Objectives

1. Understand and identify the characteristics of a logarithmic function graph.

2. Learn how to construct the graph of a logarithmic function from its mathematical expression.

3. Extract and interpret specific values from the graph of a logarithmic function.

Introduction

Duration: (10 - 15 minutes)

The purpose of this stage is to prepare students for a detailed and practical understanding of logarithmic functions, focusing on the ability to identify, construct, and interpret their graphs. This foundation is essential for students to advance to solving more complex problems and practical applications of logarithmic functions in mathematical and scientific contexts.

Context

To start the lesson on Logarithmic Function: Graph, it is important to remind students of the concept of exponential function, as logarithmic functions are the inverses of exponentials. Explain that while the exponential function grows rapidly, the logarithmic function grows more slowly but has a wide application in various fields, such as economics, biology, and technology. Use visual examples, such as graphs of exponential and logarithmic functions, to illustrate this inverse relationship and facilitate students' initial understanding.

Curiosities

An interesting curiosity is that logarithmic functions are widely used in the Richter scale, which measures the magnitude of earthquakes. This is because earthquakes vary in intensity exponentially, and the logarithm allows these variations to be represented in a more manageable way. Another example is in the pH scale used to measure the acidity or alkalinity of substances, emphasizing the importance and presence of these functions in our daily lives.

Development

Duration: (50 - 60 minutes)

The purpose of this stage is to provide students with a detailed and applied understanding of logarithmic functions, enabling them to identify the characteristics of their graphs, construct graphs from mathematical expressions, and interpret specific values from those graphs. This solid foundation is fundamental for students to tackle more complex problems and understand the practical applications of these functions.

Covered Topics

1. Definition of Logarithmic Function: Explain that the logarithmic function is the inverse of the exponential function. Detail the general form of the logarithmic function y = log_a(x), where a is the base of the logarithm and must be a positive real number different from 1. 2. Domain and Range of the Logarithmic Function: Discuss that the domain of the logarithmic function consists of all positive real numbers (x > 0) and the range is the set of real numbers (y ∈ ℝ). 3. Graph of the Logarithmic Function: Show how the graph of a logarithmic function is characterized by being a curve that grows slowly, always passing through the point (1,0) when the base is greater than 1. If the base is less than 1, the function decreases. 4. Properties of the Graph: Discuss the properties of the graph, such as the vertical asymptote at the line x = 0, the intersection with the y-axis at the point (1,0), and the behavior of the graph for values of x approaching zero and infinity. 5. Examples of Graph Construction: Provide concrete examples of how to construct graphs of logarithmic functions for different bases (e.g., log_2(x), log_10(x), log_(1/2)(x)). Show step by step how to build the graph of each of these functions. 6. Practical Applications: Briefly address some practical applications of logarithmic functions, such as in the Richter scale, pH scale, and various mathematical and scientific formulas.

Classroom Questions

1. Draw the graph of the logarithmic function y = log_2(x) and identify the intersection point with the y-axis. 2. Use the graph of the function y = log_10(x) to find the value of x when y = 2. 3. Explain how the base of the logarithm (a) influences the shape of the graph of the logarithmic function. Compare the graphs of y = log_2(x) and y = log_(1/2)(x).

Questions Discussion

Duration: (15 - 20 minutes)

The purpose of this stage is to review and consolidate students' understanding of logarithmic function graphs, ensuring they can identify, construct, and interpret these graphs. By discussing the answers in detail and engaging students with reflective questions, the teacher ensures that knowledge has been well assimilated and that students are prepared to apply these concepts in more complex situations.

Discussion

• Discussion of the Presented Questions:

• Draw the graph of the logarithmic function y = log_2(x) and identify the intersection point with the y-axis.

• • Explain that the graph of y = log_2(x) always passes through the point (1,0), as log_2(1) = 0. Additionally, the graph grows slowly for larger values of x and approaches the vertical asymptote at x = 0. Use a table of values to show additional points on the graph, such as (2,1) and (4,2).

• Use the graph of the function y = log_10(x) to find the value of x when y = 2.

• • Show that to find x when y = 2 in the function y = log_10(x), we must solve the equation 2 = log_10(x). This means that 10^2 = x, so x = 100. Demonstrate this graphically by indicating the corresponding point on the graph of y = log_10(x).

• Explain how the base of the logarithm (a) influences the shape of the graph of the logarithmic function. Compare the graphs of y = log_2(x) and y = log_(1/2)(x).

• • Detail that the base of the logarithm influences the graph's behavior. For bases greater than 1, like in y = log_2(x), the graph grows slowly. For bases between 0 and 1, like in y = log_(1/2)(x), the graph decreases. Visually compare the graphs showing how y = log_2(x) increases and y = log_(1/2)(x) decreases.

Student Engagement

1. Questions and Reflections to Engage Students: 2. How can you quickly identify if a graph represents a logarithmic function? 3. What are the main visual differences between the graphs of y = log_2(x) and y = log_(1/2)(x)? 4. Why does the logarithmic function never touch the y-axis? 5. In what practical situations do you think you can apply knowledge about logarithmic functions? 6. If the base of the logarithm were 10, how would that influence the growth rate of the graph? 7. What would be the behavior of the graph of a logarithmic function if the base were a number very close to 1?

Conclusion

Duration: (10 - 15 minutes)

The purpose of this stage is to review and consolidate students' understanding of logarithmic function graphs, ensuring they can identify, construct, and interpret these graphs. By summarizing the main points, connecting theory to practice, and discussing relevance, the teacher reinforces learning and prepares students to apply these concepts in more complex situations.

Summary

• Review the concept of the logarithmic function as the inverse of the exponential function.
• Detailed explanation of the general form of the logarithmic function y = log_a(x) and the importance of the base of the logarithm.
• Discussion on the domain and range of the logarithmic function: domain (x > 0) and range (y ∈ ℝ).
• Analysis of the behavior of the logarithmic function graph, including the increasing or decreasing curve depending on the base.
• Identification of important properties of the graph, such as the vertical asymptote at x = 0 and the intersection with the y-axis at the point (1,0).
• Practical examples of graph construction for different bases (log_2(x), log_10(x), log_(1/2)(x)).
• Discussion of practical applications of logarithmic functions, such as in the Richter scale and the pH scale.

The lesson connected theory with practice by using concrete examples of graphs and solving practical problems involving the logarithmic function. Students were able to visualize how theory translates into graphs and were guided step by step in constructing these graphs, as well as seeing real-world applications of the concept.

The topic addressed is extremely relevant to daily life, as logarithmic functions are used in various fields such as economics, biology, and technology. Understanding the logarithmic function allows for a better comprehension of natural and scientific phenomena, such as measuring the magnitude of earthquakes and determining the pH of substances, highlighting the practical importance of these concepts.