Contextualization

Mathematics is a universal language, and throughout human history, we continuously find ourselves developing new methods to describe the world around us. The invention of logarithms is a remarkable example of this continuous development. Logarithms were invented by John Napier in the 17th century to simplify complex calculations. In exact sciences and engineering, they are indispensable for solving exponential equations used to describe phenomena such as radioactive decay, population growth, chemical reactions, and much more.

A logarithm is the inverse of exponentiation and can be defined in any base, but the most common bases are 10 (common logarithm), e (natural logarithm), and 2 (binary logarithm). The logarithm of a number is the exponent to which another fixed quantity, the base, must be raised to produce that number. They are incredibly useful tools that simplify complex expressions and calculations.

Introduction

In this project, we will explore the concept of logarithms and their applications through solving logarithmic equations. A logarithmic equation is an equation that involves logarithms with the same base or different bases. Depending on the level of complexity, these equations can become a real challenge. Solving logarithmic equations involves transforming the logarithmic equation into an exponential equation to better manipulate and solve the equation.

Knowledge of basic logarithm properties is essential for solving these equations. Additionally, the ability to manipulate equations and isolate variables is a crucial skill. As we explore logarithmic equations, we will encounter fascinating concepts and properties that will help us better understand how logarithms work and how they are applied in real life.

Practical Activity

Activity Title: "Unraveling logarithmic equations through puzzles"

Project Objective

The objective of this project is to solve a series of logarithmic equation problems structured as a puzzle. Each correctly solved problem will lead to a clue, and these clues, when combined, will point to the final solution and a complete understanding of the logarithm concept and its applications.

Detailed Project Description

Each group of students will receive a set of logarithmic equation problems to solve. Each problem, once solved, will provide a clue or part of the final puzzle solution. The problems will vary in difficulty and complexity to ensure that students are effectively learning and applying the concepts learned during the project development.

The project will be divided into the following phases:

1. Research and study of the concepts necessary for solving logarithmic equations.
2. Solving logarithmic equation problems and noting the clues.
3. Combining the clues to find the final puzzle solution.
4. Project report development.

Required Materials

• Research material (books, internet, videos, etc.)
• Paper and pen
• Calculator

Detailed Step-by-Step for Activity Execution

1. First Step - Research: Start by conducting research to understand the concepts of logarithm and logarithmic equations. All group members must understand these concepts before proceeding to the next step.

2. Second Step - Problem Solving: With an understanding of the necessary concepts, the group should start solving the problems. All members should actively participate in solving the problems.

3. Third Step - Clue Combination: As they solve the problems, the group should note the clues or parts of the final puzzle solution that each problem provides. With all the clues in hand, the group should combine the clues and find the puzzle solution.

4. Fourth Step - Report: Finally, the group should prepare a project report describing the research process and problem-solving, the solution found, and the puzzle's final conclusion.

Project Deliverables

Upon completing the practical activity, each group must deliver:

1. Solutions to logarithmic equation problems.

2. The final puzzle solution.

3. The project report, containing:

   • Introduction: Should contain a general description of the project, its relevance, and the objective of this project.

   • Development: Here, the group should detail the entire project execution process, from initial research to problem-solving and the final puzzle conclusion. The theory of logarithms and logarithmic equations, the methodology used, and the discussion of the results obtained should be explained.

   • Conclusions: Should include a summary of the main points of the project, the lessons learned, and a reflection on how the concepts of logarithm and logarithmic equations were applied in practice.

   • Bibliography: Indication of all information sources used to carry out the project.