Finding The Domain Of Log₄((x²+x-20)/(x²+5x))
Introduction
In the realm of mathematics, logarithmic functions play a crucial role, particularly in solving exponential equations and modeling various real-world phenomena. Understanding the domain of a logarithmic function is fundamental to its proper application and interpretation. The domain, in simple terms, encompasses all possible input values (x-values) for which the function yields a valid output. When we delve into logarithmic expressions, we encounter certain restrictions that dictate the permissible values within the domain. These restrictions primarily stem from the inherent nature of logarithms: the argument (the expression inside the logarithm) must always be strictly positive, and the base of the logarithm must be a positive number different from 1. These conditions are paramount in maintaining the function's mathematical integrity and ensuring that the output remains a real number. This article aims to provide a comprehensive exploration of determining the domain of the logarithmic function log₄((x²+x-20)/(x²+5x)). We will embark on a step-by-step journey, dissecting the expression, identifying potential restrictions, and ultimately defining the set of all permissible x-values. By the end of this analysis, readers will gain a solid understanding of how to approach similar problems and confidently navigate the intricacies of logarithmic functions.
Understanding Logarithmic Functions
To effectively determine the domain of the given logarithmic function, let's first solidify our understanding of logarithmic functions in general. A logarithmic function is essentially the inverse of an exponential function. The expression logₐ(b) = c is equivalent to aᶜ = b, where 'a' represents the base, 'b' is the argument, and 'c' is the exponent. However, there are specific conditions that must be met for this relationship to hold true and yield a meaningful result. The base 'a' must be a positive number and cannot be equal to 1. If the base were 1, the exponential function would simply be a constant function (1 raised to any power is still 1), and its inverse would not exist. Similarly, if the base were negative or zero, the exponential function would exhibit erratic behavior, making the logarithm undefined for many values. The argument 'b' must be strictly positive. This restriction arises from the fact that no real number exponent can transform a positive base into a negative number or zero. In other words, there is no exponent 'c' that satisfies aᶜ = b when 'b' is non-positive. Therefore, the domain of any logarithmic function is fundamentally tied to these two critical conditions: the base must be positive and not equal to 1, and the argument must be positive. These constraints ensure that the logarithmic function produces real-valued outputs and maintains its mathematical consistency. Understanding these principles is crucial for tackling the problem at hand and successfully identifying the domain of log₄((x²+x-20)/(x²+5x)).
Deconstructing the Given Logarithmic Expression
Now, let's turn our attention to the specific logarithmic expression we aim to analyze: log₄((x²+x-20)/(x²+5x)). This expression presents a slightly more complex scenario than a basic logarithm due to the presence of a rational function (a fraction with polynomials in the numerator and denominator) as the argument. To determine the domain, we need to carefully consider all the restrictions imposed by both the logarithm and the rational function. Firstly, as established earlier, the argument of the logarithm, (x²+x-20)/(x²+5x), must be strictly positive. This means that the entire fraction must evaluate to a value greater than zero. Secondly, we must address any restrictions that arise from the rational function itself. Rational functions are undefined when their denominator is equal to zero, as division by zero is mathematically undefined. Therefore, we must identify any x-values that make the denominator, x²+5x, equal to zero and exclude them from the domain. In essence, we have two key conditions to satisfy: (1) the fraction (x²+x-20)/(x²+5x) > 0, and (2) the denominator x²+5x ≠ 0. These conditions interweave to define the permissible values of x. We will proceed by meticulously addressing each condition, factoring the polynomials, and identifying critical points that will help us determine the intervals where the function is valid. By systematically deconstructing the expression and considering all restrictions, we can accurately pinpoint the domain of the given logarithmic function.
Identifying Critical Points: Factoring and Solving
To solve the inequality (x²+x-20)/(x²+5x) > 0 and the equation x²+5x ≠ 0, we must first factor the polynomials involved. Factoring allows us to identify the critical points, which are the x-values where the expression can potentially change its sign or become undefined. Let's begin by factoring the quadratic in the numerator, x²+x-20. We seek two numbers that multiply to -20 and add up to 1. These numbers are 5 and -4. Thus, we can factor the numerator as (x+5)(x-4). Now, let's factor the denominator, x²+5x. We can factor out a common factor of x, resulting in x(x+5). Therefore, the factored expression becomes ((x+5)(x-4))/(x(x+5)). We can now identify the critical points. From the numerator, we have x = -5 and x = 4. From the denominator, we have x = 0 and x = -5. Notice that x = -5 appears in both the numerator and the denominator. This indicates a potential discontinuity, which we will address shortly. The critical points divide the number line into intervals, and the sign of the expression can only change at these critical points. To solve the inequality, we will analyze the sign of the expression within each interval. To ensure the denominator is not zero, we have x ≠ 0 and x ≠ -5. These restrictions are crucial for defining the domain of the rational function and, consequently, the logarithmic function. By factoring and identifying these critical points, we have laid the groundwork for determining the intervals where the logarithmic function is defined.
Analyzing Intervals and Determining the Solution Set
With the critical points identified (-5, 0, and 4), we can now analyze the intervals on the number line to determine where the expression ((x+5)(x-4))/(x(x+5)) > 0. These critical points divide the number line into four intervals: (-∞, -5), (-5, 0), (0, 4), and (4, ∞). To determine the sign of the expression within each interval, we can choose a test value within that interval and substitute it into the factored expression. In the interval (-∞, -5), let's choose x = -6. Substituting into the expression, we get ((-6+5)(-6-4))/((-6)(-6+5)) = ((-1)(-10))/((-6)(-1)) = 10/6, which is positive. Therefore, the expression is positive in the interval (-∞, -5). In the interval (-5, 0), let's choose x = -1. Substituting, we get ((-1+5)(-1-4))/((-1)(-1+5)) = ((4)(-5))/((-1)(4)) = -20/-4 = 5, which is positive. Thus, the expression is positive in the interval (-5, 0). In the interval (0, 4), let's choose x = 1. Substituting, we get ((1+5)(1-4))/((1)(1+5)) = ((6)(-3))/((1)(6)) = -18/6 = -3, which is negative. Hence, the expression is negative in the interval (0, 4). In the interval (4, ∞), let's choose x = 5. Substituting, we get ((5+5)(5-4))/((5)(5+5)) = ((10)(1))/((5)(10)) = 10/50 = 1/5, which is positive. Thus, the expression is positive in the interval (4, ∞). We are looking for intervals where the expression is greater than zero. Therefore, the solution set includes the intervals (-∞, -5), (-5, 0), and (4, ∞). However, we must remember that x ≠ -5 and x ≠ 0 due to the denominator of the rational function. The factor (x+5) appears in both the numerator and the denominator, indicating a removable discontinuity at x = -5. Although the expression is positive in the intervals immediately surrounding -5, the function is undefined at x = -5 itself. This means that -5 must be excluded from the domain. By carefully analyzing the intervals and considering the restrictions, we have identified the solution set for the inequality, which forms a crucial part of the domain of the logarithmic function.
Defining the Domain: Combining Restrictions
Now that we have analyzed the intervals and determined the solution set for the inequality (x²+x-20)/(x²+5x) > 0, we need to combine these findings with the restrictions arising from the logarithmic function and the rational function. The solution set for the inequality includes the intervals (-∞, -5), (-5, 0), and (4, ∞). This means that the expression (x²+x-20)/(x²+5x) is positive within these intervals, satisfying the fundamental requirement for the argument of a logarithm. However, we also have the restriction that the denominator of the rational function, x²+5x, cannot be equal to zero. This leads to the conditions x ≠ 0 and x ≠ -5. The critical point x = -5 warrants special attention. While the expression is positive in the intervals immediately surrounding -5, the function is undefined at x = -5 itself due to the zero in the denominator. This represents a removable discontinuity, often visualized as a "hole" in the graph of the function. Therefore, even though the interval analysis suggests that the expression is positive near -5, we must exclude -5 from the domain. Combining all these considerations, we can now define the domain of the logarithmic function log₄((x²+x-20)/(x²+5x)). The domain is the set of all x-values for which the function yields a real-valued output. Based on our analysis, this domain consists of all x-values in the intervals (-∞, -5), (-5, 0), and (4, ∞), excluding the points where the denominator is zero. Therefore, the final domain can be expressed in interval notation as (-∞, -5) ∪ (-5, 0) ∪ (4, ∞). This comprehensive domain represents the permissible input values for the logarithmic function, ensuring that the argument remains positive and the function remains well-defined. By meticulously combining the solutions from the inequality analysis with the restrictions from the rational function, we have successfully determined the domain of the given logarithmic expression.
Conclusion
In conclusion, determining the domain of a logarithmic function involving a rational expression requires a meticulous and systematic approach. We began by establishing the fundamental principles of logarithmic functions, emphasizing the crucial requirement that the argument must be strictly positive. We then deconstructed the given expression, log₄((x²+x-20)/(x²+5x)), and identified the key conditions for its validity: the rational expression (x²+x-20)/(x²+5x) must be greater than zero, and the denominator x²+5x must not be equal to zero. We proceeded by factoring the polynomials, identifying critical points, and analyzing the intervals on the number line to determine where the expression satisfies the positivity condition. We carefully considered the restrictions imposed by the denominator, particularly the removable discontinuity at x = -5. Finally, we synthesized all the information to define the domain of the function as the union of the intervals (-∞, -5), (-5, 0), and (4, ∞). This domain represents the complete set of x-values for which the logarithmic function produces a real-valued output. This comprehensive analysis highlights the importance of a thorough understanding of logarithmic functions, rational expressions, and the interplay between them. By following a step-by-step approach, we can confidently tackle similar problems and navigate the complexities of mathematical functions. The ability to accurately determine the domain of a function is not only a valuable mathematical skill but also a crucial step in applying these functions to model and solve real-world problems.
