Objectives (5 - 7 minutes)

1. Understand the concept of Logarithmic Function: The teacher should clearly and concisely explain what a logarithmic function is, how it is used, and what its main characteristics are. Students should be able to understand the definition and apply it to real problems.

2. Identify the inputs and outputs of a Logarithmic Function: The teacher should teach students how to identify the inputs (x) and outputs (y) in a logarithmic function. Students should be able to differentiate between the two and apply this knowledge in problem-solving.

3. Solve problems using Logarithmic Functions: Students should be able to apply the acquired knowledge to solve problems involving logarithmic functions. The teacher should provide practical examples and step-by-step guidance to help students practice and improve their skills.

Secondary objectives:

• Promote active student participation: The teacher should encourage active student participation during the lesson by asking questions, promoting discussions, and encouraging them to share their ideas and solutions.

• Develop critical thinking skills: Besides learning to solve problems, students should be able to analyze and evaluate different approaches to problem-solving. The teacher should encourage students to think critically and justify their answers.

Introduction (10 - 15 minutes)

1. Review of previous content: The teacher should start the lesson with a brief review of logarithms and functions concepts, as this knowledge is fundamental to understanding the logarithmic function. They can remind students about what logarithms are, how they work, and how functions are used to describe relationships between variables.

2. Problem situation 1: Sound intensity: The teacher can present the following situation: "If the intensity of a sound is measured in decibels, ranging from 0 to 120, and we want to express this variation on a logarithmic scale, how would we do that?" This situation aims to introduce the logarithmic function as a tool to represent non-linear relationships more efficiently.

3. Problem situation 2: Half-life of a radioactive element: The teacher can present the following situation: "If a radioactive element has a half-life of 100 years, how can we use a logarithmic function to determine how much of the element will remain after a certain period?" This situation aims to show students how logarithmic functions are applied in sciences like physics and chemistry.

4. Contextualization of the importance of the topic: The teacher should explain how logarithmic functions are widely used in various fields of science and engineering to model phenomena that do not follow a linear progression. They can mention real-world applications, such as measuring earthquakes (Richter scale), the acidity of a solution (pH), and population growth.

5. Curiosity 1: What is a logarithmic scale? The teacher can spark students' curiosity by explaining that a logarithmic scale is a way to represent a wide range of values on a non-linear scale. They can give examples of situations where logarithmic scales are used, such as the graph of population growth over time or the graph of an earthquake's volume relative to its intensity.

6. Curiosity 2: The origin of the logarithm The teacher can tell the story of the development of logarithms, explaining that they were invented by the Scottish mathematician John Napier in the 16th century to simplify complicated calculations. They can mention that logarithms were originally used with base 10, but nowadays, they are also used with other bases, such as base e (natural logarithm).

7. Lesson objective: Finally, the teacher should present the lesson's objective, which is to understand how logarithmic functions work, how to identify their inputs and outputs, and how to use them to solve problems.

Development (20 - 25 minutes)

1. Theory: What is a Logarithmic Function? (5 - 7 minutes) The teacher should start the theoretical part by explaining what a logarithmic function is. They should define the logarithmic function as the inverse function of the exponential function and show the relationship between logarithms and powers. The teacher should explain that the logarithmic function is written in the form y = log_b(x), where b is the base of the logarithm.

2. Theory: Characteristics of Logarithmic Functions (5 - 7 minutes) The teacher should then explain the main characteristics of logarithmic functions. They should talk about the vertical asymptote, which is a line that the function's graph approaches infinitely but never touches. The teacher should also explain that the logarithmic function has a restricted domain, meaning there are certain values of x for which the function is not defined. Additionally, the teacher should emphasize that the logarithmic function is always increasing, meaning that as x increases, y also increases.

3. Theory: Identifying the Inputs and Outputs of a Logarithmic Function (5 - 7 minutes) The teacher should then teach students to identify the inputs (x) and outputs (y) in a logarithmic function. They should show that x is the base of the logarithm and y is the value of the logarithm. The teacher can use practical examples to illustrate this concept.

4. Practice: Solving Problems with Logarithmic Functions (5 - 7 minutes) The teacher should then demonstrate how to solve problems involving logarithmic functions. They can start with simple problems and gradually increase the difficulty. The teacher should provide step-by-step guidance and explain each step of the problem-solving process.

5. Practice: Application Exercises (5 - 7 minutes) Finally, the teacher should give students the opportunity to practice what they have learned. They should provide a series of application exercises for students to solve. The teacher should circulate around the room, offering help when needed and ensuring that students are on the right track. They should encourage students to think critically and justify their answers.

Throughout the Development, the teacher should encourage active student participation by asking questions, promoting discussions, and encouraging them to share their ideas and solutions. They should also provide constructive feedback to help students improve their problem-solving skills.

Return (8 - 10 minutes)

1. Group Discussion (3 - 4 minutes): The teacher should open a group discussion to allow students to share their solutions or conclusions about the problems or exercises solved. This can be done orally or in writing, depending on the class size and available time. The teacher should encourage all students to participate and contribute their ideas and opinions. They should also ask questions to stimulate students' reflection and critical thinking.

2. Connection to Practice (3 - 4 minutes): The teacher should then ask students to reflect on how what they learned connects to the real world. They can ask questions like: "How can logarithmic functions be used to solve everyday problems?" or "In what other disciplines besides mathematics do you think logarithmic functions can be useful?" The goal of this activity is to show students the relevance of what they learned and encourage them to apply their knowledge in real situations.

3. Self-assessment (2 minutes): Finally, the teacher should ask students to self-assess their learning. They can ask students to evaluate how well they understood the concept of logarithmic function, how comfortable they feel identifying the inputs and outputs of a logarithmic function, and how confident they are in solving problems involving logarithmic functions. The teacher should remind students that self-assessment is an important learning tool as it allows them to identify their strengths and areas for improvement.

Throughout the Return, the teacher should emphasize the importance of critical thinking, reflection, and self-assessment. They should encourage students to be honest with themselves and recognize their efforts and achievements. Additionally, the teacher should provide constructive feedback to help students improve their performance and achieve their learning objectives.

Conclusion (5 - 7 minutes)

1. Summary of Content (2 - 3 minutes): The teacher should recap the main points covered during the lesson. They should highlight the concept of logarithmic function, the relationship between logarithms and powers, the characteristics of logarithmic functions, and how to identify the inputs and outputs of a logarithmic function. The teacher can give a brief summary of each topic and check if students can recall the content.

2. Connection between Theory and Practice (1 - 2 minutes): Next, the teacher should explain how the lesson connected theory, practice, and application. They should emphasize that, besides understanding the theory behind logarithmic functions, students also had the opportunity to practice problem-solving and apply their knowledge in real situations. The teacher should stress that mathematics is not just a set of rules and formulas to be memorized but a powerful tool to understand and describe the world around us.

3. Extra Study Materials (1 - 2 minutes): The teacher should suggest extra materials for students who want to deepen their understanding of logarithmic functions. These materials can include textbooks, online videos, math websites, among others. The teacher should encourage students to explore these resources on their own and use the time outside the classroom to review the content and practice more.

4. Relevance of the Topic (1 minute): Finally, the teacher should emphasize the importance of logarithmic functions in everyday life. They can mention again examples of practical applications, such as measuring earthquakes (Richter scale), the acidity of a solution (pH), and population growth. The teacher should reinforce that by understanding logarithmic functions, students are acquiring a valuable tool that can be applied in various fields of science and engineering.

Throughout the conclusion, the teacher should maintain a motivating and inspiring tone, emphasizing the importance of continuous learning and intellectual curiosity. They should express their confidence in the students and encourage them to persist, even in the face of challenges. Additionally, the teacher should reinforce that they are available to help students with any questions or difficulties that may arise.