Introduction to Logarithm

Relevance of the Topic

Logarithm is an essential concept in Mathematics. Its main function is to reverse the calculation of an exponential, allowing us to find the value of the exponent when we know the base and the result. Logarithm has applications in various areas, including science, engineering, finance, statistics, among others. Furthermore, its understanding lays the groundwork for more advanced topics, such as numerical series and differential and integral calculus.

Contextualization

Within the Mathematics curriculum, Logarithm is presented after learning about the properties of powers. It offers a new perspective on number manipulation and equation solving. The introduction to logarithm establishes the basis for understanding more complex topics that arise later, such as logarithmic and exponential functions, geometric series, among others. This is an initial milestone in the study journey of mathematics, from which students will explore increasingly deeper and challenging concepts.

Theoretical Development

Components

• Definition of Logarithm: The logarithm of a number is the power to which another value (base) must be raised to produce that number. Its definition is therefore directly related to the definition of power.

• Notation: The standard notation for representing a logarithm is log. The base with respect to which we are calculating the logarithm is written in subscript format to the right log_b(x), where b is the base and x is the value whose logarithm we are calculating. By convention, if the base is not specified, it is assumed to be 10, which is represented by the notation log(x). However, it is also common to use the base and how it changes the result of the logarithm.

• Graphical Interpretation: The logarithm of a number is equal to the exponent (the distance along the x-axis) needed to reach that number from the base, on a logarithmic scale. This interpretation provides a geometric view of the logarithm.

• Properties: Logarithms have several properties that facilitate their manipulation. Among them is the change of base property, which allows us to convert logarithms from one base to another.

Key Terms

• Logarithm: It is the number that indicates the power to which a base must be raised to produce a certain number.

• Logarithm Base: It is the number to which we are raising to obtain the desired value.

• Logarithm Argument: It is the value for which we are calculating the logarithm.

• Inversion Property: is the main property of the logarithm that differentiates it from a power. Basically, the logarithm reverses the relationship between the base and the power result.

Examples and Cases

• Example 1: the logarithm of 1000 in base 10 is 3, because 10 raised to the power of 3 results in 1000 (log_10(1000) = 3).

• Example 2: the logarithm of 8 in base 2 is 3, because 2 raised to the power of 3 results in 8 (log_2(8) = 3).

• Case 1: In solving an exponential equation, the logarithm is used to isolate the exponent. For example, in the equation 2^x=8, we can take the logarithm with base 2 on both sides to obtain x = log_2(8), which is x = 3.

• Case 2: The concept of logarithm is fundamental in computer science, where it is widely used in algorithms and algorithm performance analysis. In the case of video cards, they work with a logarithmic scale to improve the resolution and color quality displayed.

Detailed Summary

Key Points:

• Definition and Notation of Logarithm: Logarithm is an inverse function of power. In simple terms, for a certain number x, the logarithm in base b of x is the exponent to which b must be raised to produce x. This is represented by log_b(x). The base b provides the context for the calculation, and the standard base 10 is used if not specified.

• Graphical Interpretation of Logarithm: Also known as logarithmic scale, it can be visualized as the distance along the horizontal axis that gives rise to a certain value along the vertical axis. In a logarithmic scale, the increments along the horizontal axis are not constant, but multiplied by the base. This results in a graphical representation of a logarithm being a straight line, instead of the curve of an exponential.

• Logarithm Properties:

  • The change of base property is especially useful. It allows us to convert a logarithm from one base to any other. For example, to convert the logarithm of x in base a to base b, we can use the formula log_b(x) = log_a(x) / log_a(b).
  • Other important properties include addition, subtraction, multiplication, and division of logarithms.

• Examples and Cases:

  • Example 1: For the equation 10^x = 1000, we can rewrite 1000 as 10^3, resulting in 10^x = 10^3. Now, since the bases are equal, the exponents must also be equal, therefore x = 3.
  • Example 2: For the equation 2^x = 8, again we rewrite 8 as 2^3, resulting in 2^x = 2^3. Applying the same logic, we have x = 3.

Conclusions:

• Mastering the concept of logarithm and its properties is essential for the study and understanding of more advanced topics in mathematics and related areas, such as computer science and engineering.

• Logarithm provides a powerful tool for solving exponential equations, saving time and effort compared to more traditional approaches.

Exercises:

1. Exercise 1: Calculate the value of x in the equation 2^x = 16.
2. Exercise 2: Solve the equation log_2(x) = 3 for x.
3. Exercise 3: Verify if the equation log_a(b) = log_c(d) is always true, where a, b, c, and d are any positive numbers greater than 1. If true, explain why. Otherwise, provide a counterexample.