Contextualization

The logarithmic function is one of the most powerful and versatile tools that we have in mathematics, besides being the basis for diverse practical applications in various fields such as physics, economy, engineering, medicine and computing.

In this project, you will dive more deeply into the logarithmic function and its graphic representation, understanding the particularities and properties of the graph of this function. Moreover, we will understand how to determine whether a graph is or is not the graphical representation of a logarithmic function.

The logarithmic function requires a solid understanding of exponentiation, since logarithm and exponentiation are inverse operations. The graph of the logarithmic function has specific behavior, and understanding this behavior is essential to comprehend how logarithms are used in several sciences.

Logarithm has a vast range of practical applications, from the pH calculation in chemistry to the analysis of earthquake magnitude in the Richter scale, not to mention the evaluation of population growth in biology, among many other examples. In this project, we will focus on its application in finance, an area of knowledge that uses logarithms' concepts a lot.

Logarithms can be used to calculate the time required for an investment to double, making life easier for those dealing with investments and compound interest. It is no wonder that logarithms are present in various financial calculators and spreadsheets for financial calculations.

To delve into the concepts of logarithms and their applications, we recommend some reliable sources:

1. Book: "Fundamentos de Matemática Elementar" by Gelson Iezzi and Carlos Murakami. Particularly the volume 1 that deals with Logarithms.
2. Khan Academy: Logarithmic function.
3. SOS Professor: Exponential and Logarithmic Function.
4. Video: What is logarithm? - World of Mathematics.
5. Website: Brasil Escola: Logarithm.

Bear in mind that consulting these sources is just a suggestion and in no way replaces reading independently and making an individual effort in the search for knowledge.

Hands-on Activity

Activity Title: Logarithms at Action: Investments and Graphing

Project Objective:

The project's main goal is understanding and applying the concepts of logarithmic function and its graphic representation, besides the interdisciplinary nature of logarithms in financial mathematics. Students will be challenged to apply theoretical knowledge to solve real-world problems related to finances, in addition to managing execution time, working as a team, communicating effectively and solving problems creatively.

Project Description:

In groups of 3 to 5 students, you will investigate the application of logarithms in financial mathematics, particularly to calculate the time needed for an investment to double. Hence, you will create an investment scenario, define the rate of return, and calculate the time to double this investment using logarithms.

After you are done with financial mathematics, you will focus on the graphical representation of the logarithmic functions involved in the proposed problem. You will create these graphs, understand their characteristics and behavior, and learn how to use these visual representations to solve problems.

This entire process will be documented in a detailed report that will approach all project stages.

Materials Required:

• Paper and pencil
• Calculator
• Software to generate graphs (it can be a spreadsheet like Excel or software specific for graphs)
• Books and/or internet to check

Detailed Step-by-step:

1. Form groups of 3 to 5 students.
2. Each group needs to create an investment scenario. This scenario should include the initial amount to be invested and the expected rate of return (interest).
3. Based on the chosen scenario, calculate how long it will take for the investment to double using the formula for duplication time, which involves logarithms.
4. After the financial calculations, start working on the logarithmic function involved in the problem. Identify the corresponding logarithmic function from the calculated equation.
5. Plot the graph of this logarithmic function using the chosen software.
6. Analyze the graph constructed. Identify its characteristics and behavior according to the properties of logarithmic graphs you have learned in class.
7. With the gathered information, start to elaborate the project report. This report should contain: introduction, development (with the financial calculations, the logarithmic equation and the graph), conclusions and bibliography.

Project Deliverables:

The project will have two main deliverables: the graph of the logarithmic function relevant to the chosen scenario and a detailed project report.

The graph should be clean, precise and clear, showing the relevant logarithmic function. Remember, the objective of this graph is to help you visualize the properties of the logarithmic function you have studied.

The report should be detailed, well-written and properly formatted. It should follow the structure of Introduction, Development and Conclusions, with a bibliography section:

1. Introduction: Describe the context and purpose of the project and explain why the concept of logarithm is relevant and interesting.

2. Development: Describe the theory of logarithms, including the logarithmic function, its graph and the process of determining whether a graph is the graph of a logarithmic function. Describe the created investment scenario and the financial calculations performed based on this scenario. Present the graph of the logarithmic function you created and discuss its properties.

3. Conclusions: Go back to the main points of the project, summarize what you learned and discuss the conclusions you drew from the project.

4. Bibliography: List the sources you used to work on the project. These can include books, websites, videos and other resources.