LOGARITMO | FUNÇÃO LOGARÍTMICA | EXERCÍCIOS

Introduction

This tutorial is designed to simplify the concept of logarithms and logarithmic functions through practical exercises. It will cover the essential properties and operations of logarithms, making it easier for you to understand and apply these concepts in your studies.

Step 1: Understand the Basics of Logarithms

• A logarithm answers the question: "To what exponent must a base be raised to produce a given number?"
• The basic form is expressed as:

  log_b(a) = c
  

  This means that b^c = a, where:
  • b is the base
  • a is the number
  • c is the logarithm of a with base b

Practical Tips

• Remember that logarithms can only be taken of positive numbers.
• The base of a logarithm is usually a positive real number.

Step 2: Learn the Key Properties of Logarithms

Familiarize yourself with the following properties, which are fundamental in solving logarithmic equations:

1. Product Property

   • log_b(x * y) = log_b(x) + log_b(y)

2. Quotient Property

   • log_b(x / y) = log_b(x) - log_b(y)

3. Power Property

   • log_b(x^p) = p * log_b(x)

4. Change of Base Formula

   • log_b(a) = log_k(a) / log_k(b) for any positive k

Common Pitfalls

• Do not confuse the base and the argument of the logarithm.
• Ensure all inputs to logarithmic functions are positive.

Step 3: Solve Sample Exercises

Practice by working through the following types of logarithmic problems:

1. Evaluate Simple Logarithms

   • Example: Calculate log_10(100).
     • Since 10^2 = 100, then log_10(100) = 2.

2. Use Properties of Logarithms

   • Example: Simplify log_2(8) + log_2(4).
     • Using the product property: log_2(8 * 4) = log_2(32).
     • Since 2^5 = 32, log_2(32) = 5.

3. Change of Base Example

   • Convert log_10(5) to base 2.
   • Using the formula: log_2(5) = log_10(5) / log_10(2).

Real-World Application

Understanding logarithms is crucial in various fields such as science (pH scale), finance (compound interest), and computer science (algorithm complexity).

Conclusion

In this tutorial, we covered the essentials of logarithms, including their basic structure, properties, and practical exercises to enhance your understanding. Feel free to revisit the properties and practice more exercises to solidify your knowledge. Next, you may want to explore more complex logarithmic equations or applications in real-world scenarios.