Contextualization

Logarithmic functions are one of the many topics that make up the High School Mathematics discipline. Their applicability extends to various areas such as physics, chemistry, engineering, computer science, economics, among others. Understanding how logarithms work is essential to deepen your studies in these areas and even at more advanced levels of Mathematics.

Logarithms were introduced in mathematics during the 17th century, simplifying and making calculations involving very large or very small numbers more feasible. The logarithm of a number to a certain base is the exponent to which the base must be raised to obtain the number. In other words, it solves problems like: 'To which power must the base be raised to result in a given quantity?'

Introduction

Logarithmic functions, as the name suggests, use the concept of logarithm in their structure. A logarithmic function is a function in the form f(x) = log_b(x), where 'b' is the base of the logarithm and 'x' is the input of the function. The intriguing part of a logarithmic function is that it is only defined for positive inputs. Furthermore, its output is a real number for every positive input.

To better understand logarithmic functions, it is crucial to comprehend their properties. An important property is that the logarithm of a product is equal to the sum of the logarithms of the multiplied numbers. Another useful property is that the logarithm of a power is equal to the product of the exponent by the logarithm of the base.

Finally, it is relevant to know that logarithmic functions have characteristic graphs. The graph of a logarithmic function has the shape of an increasing curve that approaches the y-axis as x approaches zero.

In this project, we will explore logarithmic functions, mainly their input and output aspects, through a practical and collaborative task.

Practical Activity

Activity Title: Logarithming in the Real World

Project Objective:

The objective of this project is to provide you, students, with the opportunity to explore logarithmic functions in a practical, creative, and collaborative context. You will apply the concepts and properties of logarithmic functions to solve real-world problems and build logarithmic mathematical models.

Detailed Project Description:

Each group of 3 to 5 students will choose an area of knowledge (such as geology - to study earthquakes, chemistry - to analyze pH, economics - to understand compound interest, among others) where logarithmic functions are applied. You will research the application of logarithms in this area, how they help in understanding phenomena, and solve a problem where the logarithmic function is necessary.

You will produce a mini-podcast explaining the application, the problem-solving process, and a written presentation reporting all the details and processes of the project.

Required Materials:

• Computer with internet access for information research and podcast recording.
• Audio recording and editing software (Audacity, GarageBand, Anchor, etc.).
• Program for writing the presentation (Google Docs, Word, etc.).
• Calculator to assist in problem-solving.

Step-by-Step for Activity Execution:

Step 1: Group Formation

Form groups of three to five members. Collaborative work is important to divide research tasks, problem-solving, and material production.

Step 2: Choose the Study Area

After forming the groups, each one must choose an area of knowledge that has the application of logarithmic functions. Research and decide together.

Step 3: Research and Problem-Solving

With the chosen area, you should research more deeply how logarithmic functions are applied. Find a problem that involves the use of logarithmic functions to solve it.

Step 4: Podcast Recording

Create a script for the podcast and start recording. You should explain the application of the logarithmic function in the chosen area, detail the problem-solving process, and discuss the practical implications. Strive to make the explanation clear and accessible to listeners who are not experts.

Step 5: Document Writing

Parallel to the podcast recording, you should write the document reporting the entire project process, as specified below.

• Introduction: Provide context for the chosen area, the relevance of logarithmic functions in it, and the problem to be solved.
• Development: Explore the theory of the logarithmic function used, detail the activity carried out and the methodology used. Present and discuss the solutions found.
• Conclusion: Summarize the main points of the work, highlight the learnings obtained, and draw conclusions about the application of logarithmic functions in the chosen area.
• Bibliography: Cite all sources that assisted in the project development.

Step 6: Review and Finalization

Review both the podcast and the document to ensure they are coherent. After the review, submit both for evaluation according to the provided dates and instructions.