Objectives (5 minutes)

1. Understand the concept of a logarithmic function and its properties: Students should be able to describe what a logarithmic function is and understand its main characteristics. They need to understand the relationship between the base and the logarithm, and how it affects the graph of the function.

2. Construct graphs of logarithmic functions: Students should be able to graphically represent a logarithmic function and describe the transformations that occur when the logarithm base is changed.

3. Recognize the applications of the logarithmic function in real-world problems: Students should be able to identify real-world situations that can be modeled by a logarithmic function and interpret these models.

Secondary objectives:

• Promote interaction and collaboration among students: The lesson plan should include activities that encourage students to work together, discuss concepts, and solve problems in groups.

• Develop critical thinking and problem-solving skills: The proposed activities should stimulate students to think logically and analytically, applying the concepts learned to solve complex problems.

Introduction (10 - 15 minutes)

1. Review of previous concepts:

   • The teacher should start the lesson by reviewing the concepts of logarithms and exponential functions, as these are the basis for understanding logarithmic functions. This can be done through a brief review using simple and practical examples to reinforce students' understanding. (5 minutes)

2. Presented problem situations:

   • The teacher can propose two problem situations to introduce the topic. The first one could be: 'Imagine you are measuring the intensity of an earthquake on a Richter logarithmic scale. How would you expect the earthquake intensity to behave in relation to the distance from the epicenter?' The second situation could be: 'If you are investing money in an account that pays compound interest, how would you expect the value of your investment to behave over time?' (5 minutes)

3. Contextualization of the subject's importance:

   • The teacher should then explain that logarithmic functions are widely used in various areas, including sciences, engineering, economics, and finance. Examples such as the pH scale for measuring acidity, Weber-Fechner's law in psychology, Beer-Lambert's law in chemistry, among others, can be mentioned. The teacher should emphasize that understanding logarithmic functions is essential to comprehend and solve problems in these areas. (3 minutes)

4. Topic presentation:

   • To introduce the topic in an interesting way, the teacher can share two curiosities. The first one is that logarithms were originally developed to facilitate calculations in astronomy and navigation. The second one is that the Swiss mathematician John Napier, who invented logarithms, called them a 'wonderful device for abbreviating labor.' The teacher can then explain that although logarithms may seem complicated at first glance, they can actually simplify many complex calculations. (2 minutes)

Development (20 - 25 minutes)

1. Activity 'Building the graph' (10 - 15 minutes):

   • The teacher should divide the class into groups of up to 5 students. Each group will receive a logarithm base and several cards representing different values of x (from -5 to 5). The challenge is for them to build the corresponding graph of the logarithmic function on the whiteboard by moving the cards along the logarithm base.
   • Before starting the activity, the teacher should review the rules for building a logarithmic function graph and the transformations that occur when the logarithm base is changed. This can be done through a quick discussion and using the example of a simple logarithmic function.
   • During the activity, the teacher should circulate around the room, providing feedback and guidance as needed. They should also encourage students to discuss their strategies and explain the reasoning behind their decisions.
   • After completing the activity, each group should present their graph to the class, explaining the transformations that occurred and how the logarithm base affected the result.

2. Activity 'Applying logarithmic functions' (10 - 15 minutes):

   • For this activity, the teacher should provide the groups with a series of real-world problems that can be modeled by logarithmic functions. The problems should vary in difficulty and complexity, allowing students to apply the concepts of logarithmic functions meaningfully.
   • For example, a problem could be: 'Suppose you are measuring the concentration of a medication in the blood over time. The concentration decreases by half every 4 hours. Draw a graph showing how the medication concentration will change in the next 24 hours.'
   • Students should work together to solve the problems, discussing their ideas and strategies. The teacher should circulate around the room, offering support and guidance as needed.
   • After completing the activity, each group should present their solutions to the class, explaining how they used logarithmic functions to solve the problem.

Feedback (10 - 15 minutes)

1. Group discussion (5 - 7 minutes):

   • The teacher should gather all students and promote a group discussion. Each group will have up to 3 minutes to share the solutions or conclusions they reached during the activities.
   • During the presentations, the teacher should encourage other students to ask questions and make comments, promoting a collaborative learning environment.
   • The goal of this discussion is to allow students to see different approaches to solving the same problems, thus reinforcing the understanding of the concept of logarithmic function and its applications.

2. Learning verification (3 - 5 minutes):

   • After the discussions, the teacher should provide a general review of the main points covered, highlighting the strategies used by students to solve the problems and the difficulties encountered.
   • The teacher can ask questions to verify students' understanding, such as: 'Why does the graph of a logarithmic function never reach the x-axis?' or 'How does the change in the logarithm base affect the function's graph?'
   • The teacher should pay attention to students' responses, identifying any gaps in understanding and planning future activities to address these points.

3. Individual reflection (2 - 3 minutes):

   • To conclude the lesson, the teacher should propose that students reflect individually for a minute on what they have learned. They can ask questions like: 'What was the most important concept you learned today?' and 'What questions have not been answered yet?'
   • After the reflection, the teacher can ask some students to share their answers with the class. This not only allows the teacher to assess the effectiveness of the lesson but also helps students consolidate their learning and identify any areas that may need review or reinforcement.

4. Feedback and guidance (2 - 3 minutes):

   • Finally, the teacher should provide feedback to students, praising their efforts and highlighting the strengths of their contributions.
   • The teacher should also provide guidance on how students can continue to learn and practice the topic at home, suggesting readings, additional exercises, or relevant online resources.
   • The teacher should encourage students to ask questions, clarify doubts, and deepen their understanding. They should reinforce that learning is a continuous process and that they are available to help students in their learning journey.

Conclusion (5 - 7 minutes)

1. Summary of contents (2 - 3 minutes):

   • The teacher should recap the main points covered during the lesson, reinforcing the concept of a logarithmic function, its properties, how to construct its graph, and its applications.
   • They should emphasize the importance of understanding the transformations that occur in the graph of a logarithmic function when the logarithm base is changed, and how this knowledge can be applied to solve real-world problems.
   • The teacher can use a diagram or board to visualize these points, facilitating students' understanding.

2. Connection between theory, practice, and applications (1 - 2 minutes):

   • The teacher should highlight how the practical activities carried out during the lesson helped illustrate and deepen the understanding of theoretical concepts.
   • They can mention, for example, how the graph construction activity allowed students to visualize the transformations that occur in a logarithmic function, and how the real-world problem-solving activity demonstrated the applicability of these concepts.
   • The teacher should emphasize that mathematics is not just an abstract theory but a powerful tool to understand and solve real-world problems.

3. Extra materials (1 - 2 minutes):

   • The teacher should suggest additional study materials for students, such as math books, educational websites, explanatory videos, and online exercises.
   • For example, they could recommend reading specific chapters of a textbook, watching a video that clearly and visually explains the concept of a logarithmic function, and practicing solving more logarithmic function problems on an interactive math website.
   • The teacher should emphasize that self-study is an important part of learning and that these resources can help students consolidate and deepen their understanding.

4. Relevance of the subject (1 minute):

   • To conclude, the teacher should reaffirm the importance of understanding logarithmic functions, not only for the mathematics discipline but also for various areas of science and everyday life.
   • They can cite more examples of where logarithmic functions are used, such as in physics to describe radioactive decay, in biology to model population growth, in economics to calculate the return on an investment, among others.
   • The teacher should encourage students to continue exploring and applying what they have learned, and to realize the presence and usefulness of mathematics in their daily lives.