Introduction

Theoretical Perspective: Logarithmic inequalities are a natural extension of logarithmic equations and, therefore, a step forward in understanding logarithms and their properties.
Applicability: They are frequently found in the context of mathematical modeling, especially in exact sciences and economics, opening a door to a wide range of mathematical applications.

Previous Points: The theory of logarithms is an essential tool widely used in mathematical analysis. Therefore, before diving into logarithmic inequality, it is crucial to understand the basic concepts of logarithms, their properties, and logarithmic equations.
Subsequent Points: The study of logarithmic inequalities serves as a solid foundation for subsequent topics in advanced mathematics, such as exponential inequalities, systems of equations and inequalities, and composite functions.

Theoretical Development

Logarithmic Base: The logarithmic base is the base in which the logarithm is operated. In the study of logarithmic inequalities, we will mainly be working with logarithms of base 10 and base e.
- Base 10 logarithm: Denoted as log, it is commonly used in scientific and engineering studies.
- Natural logarithm (base e logarithm): Denoted as ln, it is widely used in differential and integral calculus.
Logarithmized Terms and Constants: In logarithmic inequalities, the terms that are subjected to the logarithm (called the argument) and the constants (such as 1, 2, 3, ...) play important roles in defining and solving the inequalities.
Properties of Logarithms: The properties of logarithms, such as the logarithm power property and the logarithm multiplication property, are essential for manipulating and simplifying logarithmic inequalities.

Inequality: It is a mathematical inequality, that is, a mathematical expression that states that two values are not equal. In logarithmic inequalities, the left and right sides of the inequality are linked by a logarithm.
Logarithmic Inequality: It is an inequality that contains logarithms. The main goal when working with logarithmic inequalities is to solve the inequality to determine the conditions under which the inequality is true.

Example 1: Solve the logarithmic inequality 2log(x+3) > log(2x-1).
- Step 1: Apply the logarithm power property to simplify the inequality.
- Step 2: Apply the logarithm multiplication property to obtain the standard form of a logarithmic inequality, i.e., "logarithm < logarithm".
- Step 3: Solve the obtained inequality using common inequality techniques.
Example 2: Determine the domain of the function f(x) = log(x+2) - 3.
- Step 1: Remember that the domain of a logarithmic function is limited by the arguments of the logarithm. That is, the argument of the logarithm must be strictly positive.
- Step 2: Solve the inequality x+2 > 0 to determine the domain of the function.

Logarithmic Inequalities: An inequality that contains one or more logarithmic expressions is called a logarithmic inequality. They can be solved using properties of logarithms and techniques for solving inequalities.
Logarithmic Base: The logarithm is the inverse of the exponential function and the logarithmic base is the base for which the logarithm is operated. In the study of logarithmic inequalities, the base and meaning of logarithms play a crucial role.
Working with Logarithmic Inequalities: The steps to solve a logarithmic inequality involve identifying the base of the logarithm, applying the properties of logarithms to simplify the inequality, transforming the inequality to the logarithmic form < logarithmic, and finally, solving the resulting inequality.
Domain of a Logarithmic Function: The domain of a logarithmic function is the set of values for which the function is defined. In the case of logarithmic functions, the domain is limited by the arguments of the logarithm that must be strictly positive.

Importance of the Study: The study of logarithmic inequalities is a natural extension of the study of logarithmic equations. They not only deepen the understanding of logarithms and logarithmic functions but are also applicable in a variety of real-world contexts.
Skills Developed: In addition to enhancing the understanding of logarithms, the study of logarithmic inequalities also develops advanced algebraic manipulation skills and logical reasoning.
Applicability: Logarithmic inequalities are a basis for understanding subsequent topics in mathematics, such as exponential inequalities, systems of equations and inequalities, and composite functions.

Exercise 1: Solve the logarithmic inequality 3 + log(x-4) < 2log(x+5).
Exercise 2: Determine the domain of the function f(x) = log(x^2 - 16).
Exercise 3: Convert the logarithmic inequality 3log3(x-2) + log3(4x+1) < 0 to the exponential form.