Equivalent Logarithmic Expressions Simplifying 2 Log(x-4) - 3 Log(x)

In the realm of mathematics, logarithmic expressions often present a fascinating challenge. These expressions, governed by the principles of logarithms, play a crucial role in various scientific and engineering disciplines. Understanding how to manipulate and simplify these expressions is paramount for success in advanced mathematical studies. Our primary focus will be on the expression 2 log (x-4) - 3 log (x), where we aim to identify the equivalent form from a set of given choices. Logarithms, at their core, are the inverse operations of exponentiation. In simpler terms, the logarithm of a number to a given base is the exponent to which the base must be raised to produce that number. For instance, the logarithm of 100 to the base 10 is 2, because 10 raised to the power of 2 equals 100. This fundamental concept underlies all logarithmic manipulations and simplifications.

Before we dive into the specific expression at hand, let’s briefly revisit some key logarithmic properties that will serve as our guiding principles. These properties are the bedrock upon which we build our understanding and manipulation of logarithmic expressions. First, we have the power rule, which states that log_b(x^p) = p log_b(x). This rule allows us to move exponents within a logarithm as coefficients and vice versa. Next, we have the product rule, which tells us that log_b(xy) = log_b(x) + log_b(y). This rule is instrumental in combining logarithms of products into sums of logarithms. Then there's the quotient rule, which states that log_b(x/y) = log_b(x) - log_b(y). This rule is essential for handling logarithms of quotients, transforming them into differences of logarithms. These properties, when applied judiciously, can help us simplify complex logarithmic expressions into more manageable forms. Understanding and mastering these rules is the key to unraveling the intricacies of logarithmic manipulations. Now, with these tools at our disposal, we are well-equipped to tackle the given expression and determine its equivalent form.

Deconstructing the Expression: 2 log (x-4) - 3 log (x)

To accurately determine the equivalent form of the given expression, 2 log (x-4) - 3 log (x), it's crucial to methodically apply the fundamental properties of logarithms we've discussed. This process will not only help us simplify the expression but also provide a deeper understanding of how logarithmic transformations work. We'll start by focusing on each term individually and then combine them using the appropriate rules. The expression consists of two main terms: 2 log (x-4) and 3 log (x). Our first step involves utilizing the power rule of logarithms. This rule is particularly useful when dealing with coefficients multiplying a logarithmic term, as it allows us to rewrite the expression with the coefficient incorporated as an exponent within the logarithm. Applying the power rule to 2 log (x-4), we can rewrite it as log ((x-4)^2). Similarly, applying the power rule to 3 log (x), we get log (x^3). This transformation is a crucial step, as it consolidates the coefficients into the logarithmic arguments, setting the stage for further simplification. By incorporating the coefficients as exponents, we've effectively transformed the original expression into a form where we can readily apply other logarithmic properties. This step not only simplifies the terms individually but also prepares the expression for the next phase of simplification, which involves combining the logarithmic terms.

Now that we've transformed the individual terms using the power rule, the expression looks like this: log ((x-4)^2) - log (x^3). The next key step is to recognize the structure of this expression. We have a difference of two logarithmic terms, which immediately suggests the application of the quotient rule. The quotient rule is a powerful tool that allows us to combine two logarithms that are being subtracted into a single logarithm of a quotient. Specifically, the quotient rule states that log_b(x) - log_b(y) = log_b(x/y). Applying this rule to our expression, we can combine the two logarithmic terms into a single logarithm. So, log ((x-4)^2) - log (x^3) becomes log (((x-4)^2) / (x^3)). This transformation significantly simplifies the expression, as we've reduced two logarithmic terms into one. The argument of the logarithm is now a fraction, where the numerator is the square of (x-4) and the denominator is x cubed. This form is much more compact and easier to analyze. The application of the quotient rule is a pivotal step in simplifying logarithmic expressions, and in this case, it leads us closer to identifying the equivalent form of the original expression.

Evaluating the Choices: A Step-by-Step Analysis

With the simplified form of the expression in hand, log (((x-4)^2) / (x^3)), we're now in a position to evaluate the given choices and determine which one is equivalent. This process involves comparing our simplified expression with each of the options, using logarithmic properties and algebraic manipulations as needed. Let’s systematically examine each choice.

Option a) $rac{2}{3} ext{log} rac{x-4}{x}$

This option presents a fraction multiplied by a logarithm of a quotient. To assess its equivalence, we need to consider whether we can transform our simplified expression into this form. The key difference lies in the argument of the logarithm and the coefficient. Our expression has (x-4)^2 in the numerator and x^3 in the denominator, whereas option a) has (x-4)/x inside the logarithm. Additionally, option a) has a coefficient of 2/3 outside the logarithm, which is not present in our simplified form. It's difficult to directly transform our expression into this form using logarithmic properties. The presence of the square in the numerator and the cube in the denominator in our expression makes it unlikely that this option is equivalent. Therefore, we can tentatively rule out option a) as a potential match.

Option b) $rac{( ext{log} (x-4))^2}{( ext{log} (x))^3}$

This option involves a ratio of logarithms raised to powers. It's crucial to recognize that this form is fundamentally different from our simplified expression, log (((x-4)^2) / (x^3)). In option b), the entire logarithm of (x-4) is squared, and the entire logarithm of x is cubed. This is in stark contrast to our expression, where we have the logarithm of ((x-4)^2) / (x^3). The logarithmic properties do not allow us to separate the logarithm of a quotient in this manner. There is no direct way to transform our simplified expression into the form presented in option b). The logarithmic operations and the algebraic operations are applied in a completely different order. Therefore, option b) is not equivalent to the original expression.

Option c) $ ext{log} rac{(x-4)^2}{x3}$

This option presents a logarithm of a quotient, which closely resembles our simplified expression: log (((x-4)^2) / (x^3)). A careful comparison reveals that the argument of the logarithm in option c) is exactly the same as the argument in our simplified expression. The numerator (x-4)^2 and the denominator x^3 match perfectly. There are no additional coefficients or manipulations needed. This direct correspondence strongly suggests that option c) is indeed equivalent to the original expression. The structural similarity and the identical arguments within the logarithm make this option a clear match. Thus, based on our analysis, option c) is the correct answer.

The Verdict: Option c) is the Equivalent Expression

After a detailed examination of each choice, our analysis conclusively points to option c) as the equivalent expression. The original expression, 2 log (x-4) - 3 log (x), when simplified using logarithmic properties, transforms into log (((x-4)^2) / (x^3)). Option c), which is log ((x-4)^2 / x^3), perfectly matches this simplified form. This equivalence is a direct result of applying the power rule and the quotient rule of logarithms.

The power rule allowed us to rewrite the coefficients as exponents within the logarithms, transforming 2 log (x-4) into log ((x-4)^2) and 3 log (x) into log (x^3). Subsequently, the quotient rule enabled us to combine the difference of two logarithms into a single logarithm of a quotient. This step transformed log ((x-4)^2) - log (x^3) into log (((x-4)^2) / (x^3)). The final expression is precisely what is presented in option c), confirming its equivalence. The other options, a) and b), do not align with this simplified form. Option a) introduces a coefficient of 2/3 and a different argument within the logarithm, while option b) presents a ratio of logarithms raised to powers, which is a fundamentally different structure.

In summary, the methodical application of logarithmic properties, coupled with a careful comparison of the simplified expression with the given choices, allows us to confidently conclude that option c) is the correct answer. This exercise highlights the importance of mastering logarithmic properties and their application in simplifying and manipulating logarithmic expressions. Understanding these concepts is not only crucial for solving mathematical problems but also for various applications in science and engineering, where logarithmic scales and transformations are frequently used.