Objectives (5 - 7 minutes)

1. Understand the concept of logarithm: The teacher must ensure that students understand what a logarithm is, how it is represented, and how it works in general terms. This can be done through simple examples, such as solving logarithmic equations and identifying patterns.

2. Apply the concept of logarithm to solve problems: After having a basic understanding of logarithm, students should be able to apply this knowledge to solve a variety of problems. They should be able to relate the logarithm of a number to the exponent to which the base must be raised to produce that number.

3. Calculate the value of a logarithm manually and using a calculator: Students should be able to calculate the value of a logarithm, both manually and using a calculator. They should understand how to perform this operation on a calculator and, more importantly, what the result means in terms of the problem they are trying to solve.

   • Secondary Objectives:
     1. Develop critical thinking and analytical skills: Solving logarithmic problems requires students to think critically and analyze the problem before applying the concept of logarithm. This will help develop their critical thinking and analytical skills.
     2. Familiarize with the calculator: The use of the calculator is not only a tool to calculate the logarithm, but also an opportunity for students to familiarize themselves with the use of the calculator, which can be useful in other areas of mathematics and other disciplines.

The objective of this stage is to clearly establish what students should learn and be able to do by the end of the lesson. This helps guide the teacher in preparing the lesson content and structuring the sequence of activities and discussions that will take place.

Introduction (10 - 15 minutes)

1. Review of previous content: The teacher should start the lesson by quickly reviewing the concepts of exponentials and logarithms. This can be done through a brief quiz or classroom discussion to check students' understanding of these topics. The teacher may ask students to recall what exponentials are, how to solve them, and how they are related to logarithms. (3 - 5 minutes)

2. Problem situation 1: The teacher can present a contextualized problem involving logarithms to spark students' interest. For example, he can ask: 'Imagine you want to calculate the amount of time it would take for a certain number of bacteria to multiply and fill a container, knowing only the bacteria's multiplication rate. How would you use logarithms to solve this problem?' This contextualized problem can help students understand the applicability of logarithms in real-world situations. (3 - 4 minutes)

3. Problem situation 2: The teacher can present another problem that requires solving a logarithmic equation. For example, he can ask: 'Given log(x) = 2, what is the value of x?' This problem can help reinforce the concept that a logarithm is the exponent to which the base must be raised to produce the number inside the logarithm. (3 - 4 minutes)

4. Contextualization: The teacher should then explain how the logarithm is a useful tool in various disciplines, including natural sciences, engineering, economics, and finance. For example, the logarithm is used to measure acidity in chemistry, calculate the intensity of an earthquake in geology, and model population growth and radioactive decay in physics. This can help show students the relevance of logarithms in their daily lives and future careers. (2 - 3 minutes)

The objective of this stage is to spark students' interest in the lesson topic by contextualizing the logarithm and showing its relevance and applicability. This can be done by presenting challenging problems that require the use of logarithms to solve, and explaining how the logarithm is used in various areas of knowledge.

Development (20 - 25 minutes)

1. Practical activity with manipulative materials: The teacher should provide students with sticks of different sizes and grid paper. Each stick represents a power of 2 (for example, the shortest stick represents 2^1, the next one 2^2, and so on). Students should arrange the sticks in ascending order of size and then draw a graph on the grid paper showing the relationship between the stick's size and the corresponding number (2^1, 2^2, and so on). Then, the teacher should introduce the idea of logarithm, explaining that the logarithm is the exponent to which the base (in this case, 2) must be raised to produce the number (the stick's size). For example, if the longest stick has a size of 32, then log_2(32) = 5, because 2^5 = 32. Students should then calculate the logarithm of various numbers represented by the sticks and record the results on the graph. This activity helps visualize the relationship between exponentials and logarithms, and understand the concept of logarithm as the 'inverse' of the exponential. (10 - 12 minutes)

2. Problem-solving activity: The teacher should present students with a series of problems that require the use of logarithms to solve. The problems should vary in difficulty level and context to keep students engaged and challenged. For example: 'Given log(x) = 3, what is the value of x?' 'If x = 2^3, what is the value of log(x)?' 'If log(x) = 2 and log(y) = 3, what is the value of log(xy)?' Students should work in groups to solve the problems, discussing among themselves and using their knowledge of logarithm to find the solutions. The teacher should circulate around the room, offering help when needed and encouraging students to explain their reasoning. (8 - 10 minutes)

3. Research and presentation activity: The teacher should ask students to research examples of how logarithm is used in real-world situations. Students should look for examples in areas such as natural sciences, engineering, economics, and finance. They should choose an example to present to the class, explaining how the logarithm is used in that situation and why it is useful. This activity helps reinforce the relevance and applicability of the logarithm, and also develops students' research, presentation, and communication skills. (5 - 7 minutes)

The objective of this stage is to allow students to explore the concept of logarithm in a practical and contextualized way, and develop their problem-solving, critical thinking, and analytical skills. The proposed activities encourage active student participation, promote collaboration and group discussion, and help consolidate the understanding of the logarithm concept.

Return (10 - 12 minutes)

1. Group discussion (3 - 4 minutes): The teacher should promote a group discussion so that students can share their solutions and conclusions from the activities carried out. Each group should present their answers to the proposed problems and the research activity. During the presentations, the teacher should encourage students to explain their reasoning and justify their answers, promoting a collaborative and argumentative learning environment.

2. Connection with theory (3 - 4 minutes): After the presentations, the teacher should summarize the main concepts discussed, highlighting how they connect with the theory presented at the beginning of the lesson. The teacher can reinforce the idea that a logarithm is the exponent to which the base must be raised to produce the number inside the logarithm, and how this was applied in the problem-solving activities. The teacher should also reinforce the idea that the logarithm is a useful tool for solving problems in various areas of knowledge.

3. Individual reflection (2 - 3 minutes): The teacher should propose that students reflect individually on what they learned in the lesson. For this, the teacher can ask questions like: 'What was the most important concept you learned today?' and 'What questions have not been answered yet?' Students should have a minute to think about these questions, and then have the opportunity to share their answers with the class. This reflection activity helps students consolidate what they have learned and identify any gaps in their understanding that may need to be addressed in future lessons.

4. Feedback and closure (2 - 3 minutes): Finally, the teacher should thank the students for their participation and effort, and provide feedback on the overall performance of the class. The teacher can praise students' strengths, such as the ability to work in teams, explain their reasoning, and apply the concept of logarithm to solve problems. The teacher can also highlight areas where students may need more practice or study, and suggest additional resources for review. The teacher should end the lesson by reminding students of the learning objectives of the lesson and encouraging them to continue practicing and studying the concept of logarithm.

The objective of this stage is to consolidate students' learning, allowing them to reflect on what they have learned and make connections between theory and practice. Group discussion, individual reflection, and teacher feedback help promote metacognition and self-assessment among students, and guide the teacher in planning future lessons.

Conclusion (5 - 7 minutes)

1. Summary of contents (2 - 3 minutes): The teacher should start the Conclusion by giving a brief summary of the main points covered in the lesson. This should include the definition of logarithm, the relationship between exponentials and logarithms, and how to calculate the value of a logarithm. The teacher should emphasize the importance of these concepts and how they apply in different contexts and disciplines.

2. Connection between theory, practice, and applications (1 - 2 minutes): Next, the teacher should highlight how the lesson connected the theory, practice, and applications of the logarithm. The teacher can recall the practical activities carried out, such as manipulating sticks to illustrate the relationship between logarithms and exponentials, and solving problems that required the use of logarithms. The teacher can also reiterate the applications of the logarithm in various disciplines and real-world situations, such as measuring acidity, modeling growth and decay, and calculating interest rates.

3. Extra materials (1 - 2 minutes): The teacher should then suggest extra materials for students who wish to deepen their understanding of the logarithm. This may include math books, educational websites, explanatory videos, and practice exercises. For example, the teacher may suggest that students explore the concept of logarithm in different bases, or practice solving more complex logarithmic equations. The teacher should emphasize that practice is essential for understanding and mastering the logarithm, and that exploring extra materials can be an effective way to reinforce learning.

4. Importance of logarithm (1 minute): Finally, the teacher should reinforce the importance of the logarithm in everyday life, highlighting that, although it may seem like an abstract concept, the logarithm is a powerful tool used in many areas of science, technology, and engineering. The teacher can give concrete examples of how the logarithm is used, such as in predicting population growth, calculating earthquake intensities, and modeling natural phenomena. This can help motivate students to continue studying and practicing the logarithm, even after the lesson has ended.

The objective of this stage is to consolidate students' learning and motivate them to continue studying and practicing the logarithm. The summary of contents, the connection between theory, practice, and applications, and the suggestion of extra materials help reinforce what students have learned and provide additional resources to deepen understanding. Emphasizing the importance of the logarithm in the real world can help maintain students' interest and motivate them to continue learning.