Equivalent Logarithmic Expressions A Step-by-Step Solution

In the realm of mathematics, particularly in logarithms, simplifying and manipulating expressions is a fundamental skill. This article aims to dissect and demystify the process of determining equivalent expressions for logarithmic functions, focusing on the specific expression: log_c((x²-1)/(5x)). We will explore the properties of logarithms, apply them step-by-step, and identify the equivalent expression from a given set of options. Understanding these principles is crucial not only for solving mathematical problems but also for grasping the broader applications of logarithms in various scientific and engineering fields.

Understanding Logarithms and Their Properties

Before we dive into the problem, let's establish a solid understanding of logarithms and their essential properties. A logarithm is essentially the inverse operation of exponentiation. In simpler terms, if we have an equation b^y = x, the logarithm of x to the base b is y, written as log_b(x) = y. Here, b is the base, x is the argument, and y is the exponent.

Logarithms possess several key properties that enable us to manipulate and simplify expressions. These properties are the bedrock of logarithmic operations and are indispensable for solving complex problems. Let's delve into the fundamental properties that we will utilize in dissecting the given expression. The product rule of logarithms states that the logarithm of the product of two numbers is equal to the sum of the logarithms of the individual numbers. Mathematically, this is expressed as: log_b(mn) = log_b(m) + log_b(n). This property is incredibly useful when dealing with products within logarithmic expressions, allowing us to break them down into simpler terms. The quotient rule of logarithms asserts that the logarithm of the quotient of two numbers is equal to the difference between the logarithms of the numerator and the denominator. The mathematical representation of this rule is: log_b(m/n) = log_b(m) - log_b(n). This property is invaluable for handling fractions within logarithmic expressions, enabling us to separate the numerator and denominator into distinct logarithmic terms. The power rule of logarithms stipulates that the logarithm of a number raised to a power is equal to the product of the power and the logarithm of the number. Symbolically, this is represented as: log_b(m^p) = plog_b(m)*. This rule is particularly useful when dealing with exponents within logarithmic expressions, allowing us to bring the exponent outside the logarithm as a coefficient. Mastering these properties is not merely about memorizing formulas; it's about understanding their application and how they transform logarithmic expressions. With these properties at our disposal, we can confidently tackle the task of simplifying and identifying equivalent expressions for the given logarithmic function.

Deconstructing the Expression: log_c((x^2-1)/(5x))

Now, let's apply these principles to our given expression: log_c((x²-1)/(5x)). The first step is to recognize that we have a quotient inside the logarithm. The quotient rule of logarithms comes into play here, allowing us to separate the numerator and the denominator. Applying the quotient rule, we get: log_c(x²-1) - log_c(5x). This transformation is a critical step in simplifying the expression, as it breaks down the complex fraction into two separate logarithmic terms. Next, we focus on the term log_c(x²-1). We can further factor the expression x²-1 as a difference of squares: (x-1)(x+1). This factorization is a common algebraic technique that simplifies many expressions. Substituting this back into our logarithmic term, we have: log_c((x-1)(x+1)). Now, we can apply the product rule to this term, which states that the logarithm of a product is the sum of the logarithms. Applying the product rule, we get: log_c(x-1) + log_c(x+1). This expansion provides a more detailed representation of the original term, highlighting the individual factors. Shifting our attention to the second term in our expression, log_c(5x), we again encounter a product within the logarithm. Applying the product rule here, we get: log_c(5) + log_c(x). This decomposition separates the constant and variable components of the term, making it easier to analyze. Now, let's combine all the transformations we've made. Our original expression log_c((x²-1)/(5x)) has been broken down into: log_c(x-1) + log_c(x+1) - (log_c(5) + log_c(x)). Distributing the negative sign, we get: log_c(x-1) + log_c(x+1) - log_c(5) - log_c(x). This final form represents the fully expanded and simplified version of the original logarithmic expression. By systematically applying the properties of logarithms and algebraic factorization, we have successfully deconstructed the complex expression into its fundamental components. This step-by-step approach is crucial for understanding the transformations and identifying equivalent expressions.

Identifying the Equivalent Expression

Now that we have fully deconstructed the original expression, log_c((x²-1)/(5x)), into its simplified form: log_c(x-1) + log_c(x+1) - log_c(5) - log_c(x), we can compare this result with the given options to identify the equivalent expression. This process involves matching the simplified form with the options provided, ensuring that all terms and their signs align correctly. Let's consider the options one by one and see how they compare to our simplified expression.

Option 1: log_c(x²) - log_c(5x) - 1. This option appears to be an attempt to apply the quotient rule initially, but it deviates in the subsequent steps. The term log_c(x²) can be further simplified using the power rule to 2log_c(x). However, this does not account for the factorization of x²-1 and the resulting logarithmic terms. Additionally, the “-1” term is not present in our simplified expression, making this option incorrect. Therefore, Option 1 does not match our derived simplified form, and we can confidently rule it out as the equivalent expression.

Option 2: 2log_c(x) - (log_c(5) + log_c(x)) - 1. This option incorporates the power rule for logarithms by expressing log_c(x²) as 2log_c(x) and correctly applies the product rule to expand log_c(5x) into log_c(5) + log_c(x). However, similar to Option 1, the factorization of x²-1 is missing, and the additional “-1” term does not align with our simplified expression. This discrepancy indicates that Option 2 is not equivalent to the original expression. While the initial steps of simplification are accurate, the absence of terms derived from the factorization of x²-1 and the presence of an extraneous constant term make it an incorrect match.

Option 3: log_c(x²-1) - (log_c(5) + log_c(x)). This option correctly applies the quotient rule in the initial step, separating the numerator and denominator of the original expression. It also accurately expands log_c(5x) using the product rule to log_c(5) + log_c(x). Furthermore, this option retains the log_c(x²-1) term, which can be further expanded to log_c(x-1) + log_c(x+1), matching our derived simplified form. Upon comparing Option 3 with our simplified expression, we find a perfect match: log_c(x-1) + log_c(x+1) - log_c(5) - log_c(x). This confirms that Option 3 is indeed the equivalent expression for the given logarithmic function. The careful application of logarithmic properties and algebraic factorization has led us to the correct answer, demonstrating the importance of a systematic approach in solving mathematical problems.

Conclusion

In conclusion, we have successfully identified the expression equivalent to log_c((x²-1)/(5x)) by systematically applying the properties of logarithms and algebraic factorization. The correct equivalent expression is log_c(x²-1) - (log_c(5) + log_c(x)). This exercise underscores the importance of understanding and applying logarithmic properties, such as the product, quotient, and power rules, in simplifying and manipulating logarithmic expressions. Mastering these concepts is crucial for success in mathematics and related fields, enabling us to tackle complex problems with confidence and precision. By breaking down the expression step-by-step, we were able to transform it into its simplest form and accurately identify the equivalent expression from the given options. This methodical approach not only helps in solving specific problems but also enhances our overall problem-solving skills in mathematics.

Which of the following expressions is equivalent to log_c((x²-1)/(5x))?

Equivalent Logarithmic Expressions A Step-by-Step Solution