Contextualization

The logarithmic function is one of the most important functions in mathematics and is the inverse of the exponential function. It has an exceptional property: the ability to transform multiplications into additions, making complex calculations easier to handle. The logarithm plays a fundamental role in many areas of mathematics, including algebra, calculus, trigonometry, geometry, number theory, and even in probability theory.

The logarithmic function may seem very abstract at first glance, but it has many practical applications. It is used in many scientific and technological fields, such as physics, engineering, chemistry, biology, computer science, and even in economic and social areas. In physics, for example, the Richter scale, which measures the intensity of earthquakes, is based on logarithms. In chemistry, the concept of pH, which measures the acidity or alkalinity of a solution, is also based on logarithms. In the field of economics, logarithms are used to understand the exponential growth of money over time.

The main objective of this project is to understand the logarithmic function and its practical applications. We will explore the concept of logarithms, their properties, how to solve them in different bases, and how they are used in real life. This project goes beyond mathematics, also involving physics and chemistry, to give an idea of how logarithms are applied in these disciplines.

Practical Activity

Activity Title: "Logarithm on Earth: Tremors, pH, and Finances"

Project Objective:

The objective of the project is to apply the understanding of logarithmic functions in practical situations, in order to understand how these functions are present in our daily lives and in various areas of knowledge. This project also aims to develop research skills, teamwork, communication, and scientific reporting.

Detailed Project Description:

1. Research and Planning Phase: In this phase, each team should organize to study and research the logarithmic function, understand the concept, properties, and practical applications. Based on the research, the team should create a plan for the project development.

2. Implementation Phase: Each team will develop three independent mini-projects, each focusing on a practical application of the logarithmic function, namely: Richter Scale (Physics), pH Concept (Chemistry), and Exponential Growth of Money (Economics/Financial Mathematics).

   • In the first mini-project, students should create a model for the Richter scale, which is logarithmic, and use this model to calculate the magnitude of hypothetical earthquakes.
   • In the second mini-project, students will explore the concept of pH, which is a logarithmic measure of the acidity or alkalinity of a solution. They should conduct experiments with different substances, measure their pH levels, and use logarithms to understand and interpret the results.
   • In the third mini-project, students will use logarithms to understand the exponential growth of money. They will create and solve problems of financial mathematics involving compound interest, using logarithms.

3. Reporting Phase: In this phase, each team should write a detailed report on the project. Each mini-project should be reported separately, including the theoretical concept involved, the implementation methodology, the results obtained, and the interpretation of these results. The team should also include an overall conclusion of the project, describing the learnings obtained and how the understanding of logarithmic functions was deepened through the project.

Required Materials:

• Internet access for research;
• Paper and writing material for notes and calculations;
• Basic chemistry laboratory equipment for the pH experiment (e.g., pH indicator, various substances for testing) - optional, can be theoretical only;
• Mathematical calculation software or spreadsheets for financial calculations.

Detailed Step-by-Step for Activity Execution:

1. Organize into groups of 3 to 5 students. Each group should appoint a leader to manage the organization of work and communication with the teacher.

2. Start with the research and planning phase. Research logarithmic functions, their properties, and applicability. Use the provided resources and look for other means as well.

3. Discuss as a team and develop an implementation plan for each of the three mini-projects. Set goals, task division, and a schedule for project development.

4. Implement each mini-project according to the plan established in the planning phase.

5. At the end of each mini-project, analyze and discuss the results obtained. How were logarithms used? How did they help understand the situation or problem?

6. After completing the three mini-projects, start the reporting phase. Write a report detailing the development of the mini-projects, the applied logarithmic concepts, the procedures performed, the results obtained, and the conclusions reached.

7. The final part of the report should be an overall conclusion of the project. Express the learnings acquired, the importance of logarithms in different fields, and how the project helped deepen the understanding of the logarithm concept.

8. Finally, review the report, check if all necessary aspects are covered, and if the information is presented clearly and organized. Submit the report to the teacher.

Project Delivery:

The final project delivery will consist of a complete report detailing the execution of the three mini-projects. This document should include the following sections: Introduction (contextualization of the theme and project objective), Development (detailed description of the three mini-projects, with theoretical explanations, procedures performed, results achieved, and discussion of these results), Conclusion (review of the main points, learnings, and conclusions about the project), and Bibliography (references used for the project execution).

This report serves as a complete record of your work and will be used to assess your understanding of the logarithm concept, the ability to apply this concept, teamwork skills, and scientific communication abilities.