Contextualization

Logarithm is a mathematical operation created in the 17th century by John Napier, representing the inverse of exponentiation, that is, the logarithm of a number in a certain base is the exponent to which the base must be raised to obtain the number. Logarithm brought great ease in performing complex operations, particularly in the areas of multiplication, division, exponentiation, and rooting, transforming these operations into additions, subtractions, and simple multiplications or divisions, respectively.

Particularly when applied to equations, logarithm has the power to transform an exponential equation into a linear one, which simplifies problem-solving immensely. That's why logarithm is extremely useful and recurrent in sciences like physics and chemistry, as well as in applied disciplines like engineering and computer science.

Importance

The importance of logarithm is evident in various fields of science and technology. In physics, radioactive decay, an inherent phenomenon in certain elements, is modeled by a logarithmic function, as well as the earthquake magnitude scale. In computer science, search and sorting algorithms are often logarithmic in their time or space complexity, indicating how quickly the execution time or memory usage increases as the input size grows. In economics, the term 'exponential growth' is often used to describe logarithmic growth.

Furthermore, understanding how to solve logarithmic equations is a crucial step for students of mathematics, physics, chemistry, engineering, and computer science, as these equations are frequently involved in problem-solving in these areas.

Practical Activity

Activity Title: Logarithms in the Real World

Project Objective

Explore and understand how logarithms are used in real life by practicing the resolution of logarithmic equations.

Detailed Project Description

Divided into groups of three to five students, the task is to identify a real phenomenon that can be described by a logarithmic equation. Some examples include: population growth, radioactive decay, earthquake magnitude, analysis of computational algorithms, pH calculation, sound (decibels), among others.

The choice of the phenomenon should be motivated by initial research, which must be described in the final report (see more below in the Project Deliveries section), and approved by the teacher.

After selecting and approving the phenomenon to be studied, students should search for real data/studies related to the chosen phenomenon, with the aim of modeling such phenomenon through a logarithmic equation. These data can be taken from books, scientific articles, reliable websites, among others. The source must be referenced and described in the final report.

Based on the collected data, students should establish and solve one or more logarithmic equations that represent the phenomenon under study, demonstrating the usefulness of logarithms in practical contexts.

Required Materials

• Notebook for notes
• Pens and pencils
• Calculator
• Internet access for research
• Mathematics / Science books (if available in the school library)

Detailed Step-by-Step

1. Formation of groups of 3 to 5 students.
2. Research on phenomena that can be described by logarithmic equations.
3. Discussion and selection of the phenomenon to be studied in the group.
4. Presentation to the teacher of the chosen phenomenon for approval.
5. Research on real data/studies related to the chosen phenomenon.
6. Based on the collected data, development of one or more logarithmic equations describing the phenomenon.
7. Resolution of logarithmic equations.
8. Writing the final report describing the entire process, from initial research to equation resolution.

Project Deliveries

At the end of the project, each group must submit a written report containing:

1. Introduction: In this section, students must contextualize the chosen phenomenon, explain its relevance and application in the real world, and also describe the project's objective.

2. Development: In this section, students must present the theory of logarithms and logarithmic equations, explain in detail the activity carried out, the methodology used, and finally present and discuss the results obtained.

3. Conclusion: Summarizing the main points, students must explain what they learned from the project, the skills developed, and the conclusions about the study conducted.

4. Bibliography: Students must list all sources used for the project, such as books, web pages, videos, etc.

The report should demonstrate not only the knowledge acquired in logarithms and logarithmic equations but also the socio-emotional skills developed, such as time management, communication, problem-solving, creative thinking, and proactivity.