Contextualization

Introduction

The logarithmic function is one of the most important tools in mathematics, with a wide range of applications in various disciplines such as physics, engineering, computer science, among others. The term 'logarithm' was introduced by John Napier in the 17th century and is derived from the Greek words 'logos' (ratio) and 'arithmos' (number). Logarithmic functions have unique properties that make them valuable instruments in solving complex problems.

Logarithms are operations that are inverse to exponentiation, that is, if a^b = c, then log_a(c) = b. Therefore, the logarithmic function is the inverse of the exponential function. To understand it, it is essential to have a good understanding of the exponential function.

Moreover, the logarithmic function is one of the main tools for dealing with very large or very small numbers. In many situations, it is easier to work with the logarithm of a number than with the number itself. This fact makes the logarithmic function an essential component in various areas of mathematics and its applications.

Importance and Applications

The logarithmic function plays a crucial role in many areas of science. In physics, logarithms are used to express the magnitude of phenomena ranging from micro to macro scales, such as the Bohr Radius (the smallest length in quantum physics) and the distance between galaxies. In biology, they are used to calculate pH - a measure of the acidity or alkalinity of a solution. In computer science, they are used in algorithms for data processing and complex calculations.

In the financial world, logarithmic functions are used to calculate the time needed for an investment to double (the 'Rule of 72') and to convert compound interest rates to equivalent simple rates.

They are also used in the calculation of radioactive decay, population growth, earthquakes (to calculate the magnitude on the Richter Scale), psychophysics (to calculate the perception of intensity of sensory stimuli), among many other examples.

Practical Activity

Activity Title: Logarithm in Everyday Life

Project Objective

The objective of this project is to connect the theoretical knowledge of the logarithmic function with practical applications in everyday life, allowing for the understanding of the concept and the improvement of technical and socio-emotional skills.

Detailed Project Description

Groups should choose a topic where the logarithmic function is applied. From there, they should develop a mathematical model for a real problem within that topic.

Possible topics include:

1. Finance: How does time affect the growth of money in a savings account?
2. Chemistry: How is the pH of a solution calculated?
3. Geography/Physics: How is the intensity of an earthquake measured?

The choice of topic should take into account the importance of the logarithm in its description and the relevance of the theme to the students' reality.

Required Materials

• Computer with internet access for research
• Text editing software for report writing
• Chart creation software or spreadsheet (e.g. Excel, Google Sheets)

Step by Step

1. Group Formation: Groups should consist of 3 to 5 students.
2. Topic Selection: Each group should choose a topic for the practical application of logarithms.
3. Research: Students should research on the internet, books, and other resources the application of logarithms in their topic, as well as the theory necessary to understand this application.
4. Model Creation: Students should create a mathematical model that represents the chosen system or phenomenon.
5. Problem Solving: Students should use the created model to solve one or more real problems within the chosen topic.
6. Report Writing: Each group should prepare a report detailing the entire process. The report should include: Introduction, Development, Conclusions, and Bibliography used.
7. Presentation: Each group should make a presentation to the class, explaining the work done and what they have learned.

Project Deliverables

The final product of the project should include:

1. Mathematical Model: Detailed description of the model, including variables, equations, and the application of the logarithmic function.

2. Problem Solving: Description and resolution of the proposed problems using the model.

3. Report: The report should be divided into four parts:

   • Introduction: Description of the chosen topic and the objective of the work.

   • Development: Detailed description of the mathematical model and the problems solved.

   • Conclusions: Reflection on what was learned, the difficulties encountered, and suggestions for future work.

   • Bibliography: List of materials used for the project development.

4. Presentation: An oral presentation detailing the project to the class.

The written documentation should be a reflection of what was developed in the activity. Therefore, it is essential that it demonstrates the group's understanding of the topic and the application of logarithmic functions in a practical context.