Project: The Logarithmic Scale

Contextualization

A logarithm is a mathematical operation that is the inverse of exponentiation. If a number $a$ can be expressed as a power of another number $b$, then the logarithm of $a$ to the base $b$ is equal to that exponent. Logarithms have numerous practical applications, especially in the field of science and engineering.

Theoretical Introduction

In the 17th century, mathematics began to develop rapidly, especially trigonometry and algebra. In this context, John Napier, a Scottish mathematician, introduced the concept of logarithm, which was conceived as a means to simplify complex calculations, especially multiplication and division. Napier constructed a table of logarithms that reduced the task of multiplying and dividing to the task of adding and subtracting, respectively. The discovery was revolutionary and greatly facilitated the mathematical calculations of the time.

Logarithms are also intrinsic to solving exponential equations, making them fundamental to the study of exponential functions and to understanding the concepts of exponential growth and decay. Another application of logarithms in mathematics is to solve differential equations, an area of study that plays a role in many branches of science and engineering.

Contextualization

Logarithms have practical applications in several areas of knowledge. In physics and engineering, for example, logarithms are used to solve problems involving radioactive decay and electric circuits. In biology, they are used to model populations of organisms. In economics, logarithms are often used in financial analysis to calculate rates of return. In computing science, logarithms play a crucial role in search and sorting algorithms.

The importance of the logarithm goes beyond the field of mathematics and appears in everyday situations even if we do not realize it. For example, the Richter scale, which measures the magnitude of earthquakes, and the decibel scale, used to measure sound intensity, are both based on logarithms.

To learn more about the subject, you can consult the following references:

• Book "Matemática para o Ensino Médio - Volume Único", by Gelson Iezzi, Osvaldo Dolce and David Degenszajn.

• Logarithm article on Wikipedia

• Khan Academy Logarithm Course

Practical Activity: The Logarithmic Scale

Project Objective

The objective of the project is to develop students' understanding of the concept of logarithms, their nature and application, enabling them to transform a set of linear data into a logarithmic scale and vice versa. The students will also experience the use of logarithms in real-world situations related to intensity measurements, such as sound (decibels) and earthquakes (Richter scale).

Detailed Project Description

The students will be divided into groups of 3 to 5 people. Each group will be responsible for collecting a set of data that can be converted to a logarithmic scale and performing the conversion.

Examples of data that can be collected include: word frequencies in a text, number of hits on a website, sound intensities, etc. This data should initially be collected and plotted on a linear scale.

Afterwards, the students should convert the data to a logarithmic scale and recreate the graphs. The students should be able to explain how and why the shape of the graph has changed and what this indicates about the data.

Materials Required

• Computer with internet access

• Google Sheets or Excel

• Calculator (optional)

Detailed Step-by-Step

1. Form groups of 3 to 5 students.

2. Choose an area of interest for data collection. It could be something related to everyday life, the school environment or even online data available on websites.

3. Collect the data and enter it into a table in Google Sheets or Excel.

4. Plot this data on a linear scale.

5. Use the spreadsheet's logarithm function to convert the data to a logarithmic scale.

6. Recreate the graph with the logarithmized data.

7. Analyze the generated graphs. How have the shapes changed? What does this mean?

8. Prepare a presentation explaining the data collection, the transformation to the logarithmic scale, and the interpretation of the graphs.

9. Write a detailed report on the entire process and findings.

Deliverables and Written Document

Each group must deliver:

1. A PowerPoint presentation or similar explaining the entire process carried out.

2. The original and transformed data set.

3. The graphs generated before and after the transformation to the logarithmic scale.

4. A written report including:

• Introduction: The relevance and application of logarithms in the real world and the objective of this project.

• Development: The theory behind the logarithm, the detailed explanation of the process of collecting the data, graphing, transforming to the logarithmic scale and re-graphing. The methodology used and the results obtained should also be presented.

• Conclusion: The main points of the project, the lessons learned and the conclusions drawn.

• Bibliography: All sources of learning and consultation for the execution of the project.