Context

Logarithms are a topic deeply connected to Mathematics, being a key concept in various areas of knowledge. They find applications ranging from Computer Science, in algorithm complexity, to Biology, in models of population growth and decay. In the specific realm of Mathematics, logarithms and inequalities become essential tools in solving exponential equations or studying function variations.

Inequalities with logarithms represent an even more specific and useful application of the concept of logarithms. This project aims to understand and apply logarithmic inequalities, a competence indispensable especially for advanced studies in Mathematics and its related sciences.

Logarithms are an integral part of many systems we use daily, and whose operation largely depends on the use of this mathematics. Understanding the principles and applications of logarithmic inequalities can improve our understanding of how the systems around us operate and behave.

The project proposed in this document aims to introduce students to the resolution of inequalities with logarithms, exploring the topic in a theoretical and practical way, fostering teamwork and encouraging research and exploration of various sources of knowledge.

For a solid foundation and further understanding of the subject, the following reliable references can be used:

1. Book “Fundamentos de Matemática Elementar: Logaritmos” by authors Gelson Iezzi and Carlos Murakami.
2. Khan Academy - Introduction to Logarithmic Inequality
3. YouTube Channel - Resolution of Logarithmic Inequalities.

Practical Activity

Activity Title: 'Unraveling Logarithmic Mysteries'

Project Objective

The project's objective is to solve a set of real-world problems based on logarithmic inequalities and present the solutions and resolution processes in a creative way, demonstrating understanding of logarithmic concepts and teamwork.

Detailed Project Description

Students will be divided into groups of 3 to 5, and each group will be presented with a real-world problem involving the use of logarithmic inequalities. The problems will be complex and will require collaboration and team discussion to solve. Each group must, at the end of the project, present a detailed report with the problem solution, the theory used to reach that solution, and an oral presentation illustrating the resolution process and final conclusions.

Required Materials

1. Paper and pencil for drafting.
2. Calculator with logarithmic functions.
3. Computer with internet access for research.
4. Presentation software (e.g., PowerPoint, Google Slides) to create the presentation.
5. Camera to film the presentation (can be a cellphone camera).

Detailed Step-by-Step for Activity Execution

1. Form groups of 3 to 5 students and assign each group a real-world problem involving logarithmic inequalities.
2. Each group should initially discuss and understand the problem, identifying where the logarithmic inequality fits in.
3. Students should research, individually and in groups, to understand the theoretical concepts needed to solve the problem.
4. After the research, students should gather and start solving the problem, collaborating and discussing the process steps.
5. After solving the problem, students should meet to discuss the conclusions and how they will present their findings.
6. The group will prepare a detailed written report, including the Introduction (contextualization and relevance of the real-life problem), Development (discussion of the theory and processes), Conclusion (learnings and insights generated by the problem), and Bibliography.
7. Simultaneously, the group will create a visual presentation to illustrate the process, challenges, discoveries, and conclusions.
8. The group will film the presentation and submit the video along with the written report.

Project Deliverables

At the end of the project, each group must deliver:

1. A written report containing:

• Introduction: Contextualization and relevance of the proposed real-life problem.
• Development: In-depth discussion of the theory used, detailing the problem resolution and the methodology employed.
• Conclusion: Reflections and learnings generated during the problem resolution.
• Bibliography: Listing and description of the consultation sources used for project development.

2. A video of the group's oral presentation, demonstrating the problem resolution and the conclusions reached.

This project will allow students to delve into the theory of logarithmic inequalities, apply them in solving real-world problems, work in teams, and develop skills in research, problem-solving, oral presentation, and technical writing.