Equivalent Expression Of 2 Log(x-4)-3 Log(x) A Step-by-Step Solution

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In the realm of mathematics, logarithmic expressions play a pivotal role in simplifying complex calculations and revealing underlying relationships between variables. Logarithms serve as the inverse operation to exponentiation, providing a powerful tool for manipulating equations and solving for unknown quantities. In this comprehensive article, we delve into the intricacies of logarithmic expressions, focusing on transforming and simplifying them using fundamental logarithmic properties. Our exploration centers around the expression $2 \log (x-4)-3 \log (x)$, where we aim to determine its equivalent form from a set of given choices. The journey through this mathematical landscape will not only solidify your understanding of logarithmic manipulations but also enhance your problem-solving skills in algebra and beyond. This introduction sets the stage for a detailed investigation into logarithmic transformations, equipping you with the knowledge to confidently tackle similar challenges in your mathematical pursuits.

Before we embark on the transformation of the given logarithmic expression, it's essential to solidify our understanding of the fundamental properties that govern logarithmic operations. These properties act as the building blocks for manipulating logarithmic expressions, enabling us to simplify complex forms and reveal underlying relationships. Let's explore these key properties:

  1. Power Rule: This rule states that the logarithm of a number raised to a power is equal to the product of the power and the logarithm of the number. Mathematically, it's expressed as $\log_b(a^c) = c \log_b(a)$, where b is the base of the logarithm, a is the number, and c is the power. The power rule is instrumental in moving exponents within logarithmic expressions, either bringing them down as coefficients or incorporating them into the argument of the logarithm.
  2. Product Rule: The logarithm of the product of two numbers is equal to the sum of the logarithms of the individual numbers. Formally, it's written as $\log_b(mn) = \log_b(m) + \log_b(n)$, where b is the base, and m and n are the numbers being multiplied. The product rule allows us to decompose the logarithm of a product into simpler logarithmic terms, facilitating simplification and analysis.
  3. Quotient Rule: Conversely, the logarithm of the quotient of two numbers is equal to the difference of the logarithms of the individual numbers. Expressed mathematically, we have $\log_b(\frac{m}{n}) = \log_b(m) - \log_b(n)$, where b is the base, and m and n are the numerator and denominator, respectively. The quotient rule provides a means to separate the logarithm of a fraction into distinct logarithmic components, aiding in simplification and problem-solving.

These logarithmic properties serve as the foundation for manipulating and simplifying logarithmic expressions. By mastering these rules, we can effectively transform complex expressions into more manageable forms, paving the way for solving equations, analyzing functions, and tackling a wide range of mathematical problems. In the subsequent sections, we'll apply these properties to the expression at hand, unraveling its equivalent form step by step.

Now, let's embark on the journey of transforming the given expression: $2 \log (x-4)-3 \log (x)$. Our goal is to manipulate this expression using the logarithmic properties we've discussed, ultimately arriving at its equivalent form. We'll proceed step by step, applying the properties in a strategic manner to achieve simplification.

Step 1: Applying the Power Rule

Our first step involves applying the power rule to both terms in the expression. Recall that the power rule states $\log_b(a^c) = c \log_b(a)$. We can use this rule in reverse to move the coefficients 2 and 3 inside the logarithms as exponents. Applying this to our expression, we get:

2log⁡(x−4)−3log⁡(x)=log⁡((x−4)2)−log⁡(x3)2 \log (x-4)-3 \log (x) = \log ((x-4)^2) - \log (x^3)

In this step, we've transformed the coefficients into exponents, setting the stage for further simplification using the quotient rule.

Step 2: Applying the Quotient Rule

Next, we'll leverage the quotient rule, which states $\log_b(\frac{m}{n}) = \log_b(m) - \log_b(n)$. We have a difference of two logarithms, which perfectly matches the form required for applying the quotient rule. Applying this rule, we combine the two logarithmic terms into a single logarithm of a quotient:

log⁡((x−4)2)−log⁡(x3)=log⁡((x−4)2x3)\log ((x-4)^2) - \log (x^3) = \log \left(\frac{(x-4)^2}{x^3}\right)

At this juncture, we've successfully consolidated the expression into a single logarithmic term. The argument of the logarithm is a fraction, which represents the quotient of the two original logarithmic terms. This transformation provides a more compact and insightful representation of the expression.

Step 3: Expanding the Square (Optional)

While the expression is now in a simplified logarithmic form, we can optionally expand the square in the numerator to gain further insight into the expression's structure. Expanding $(x-4)^2$ gives us $x^2 - 8x + 16$. Substituting this back into the expression, we get:

log⁡((x−4)2x3)=log⁡(x2−8x+16x3)\log \left(\frac{(x-4)^2}{x^3}\right) = \log \left(\frac{x^2 - 8x + 16}{x^3}\right)

This step provides an alternative representation of the expression, showcasing the polynomial structure within the logarithm's argument. Whether this expansion is necessary depends on the specific context and the desired level of detail.

Through these steps, we've effectively transformed the original expression using the power and quotient rules of logarithms. The final form, $\log \left(\frac{(x-4)2}{x3}\right)$, represents the equivalent expression in a simplified and insightful manner. This transformation highlights the power of logarithmic properties in manipulating and simplifying complex mathematical expressions.

Now that we've successfully transformed the given expression, let's analyze the provided choices to determine which one matches our simplified form. The original question presents the following options:

a) $\frac{2}{3} \log \left(\frac{x-4}{x}\right)$ b) $\frac{(\log (x-4))^2}{(\log (x))^3}$ c) $\log \left(\frac{(x-4)2}{x3}\right)$ d) None of the above

Our transformed expression is $\log \left(\frac{(x-4)2}{x3}\right)$. Comparing this to the choices, we can readily identify the equivalent option.

Choice a: $\frac{2}{3} \log \left(\frac{x-4}{x}\right)$ This choice does not match our transformed expression. The coefficient $\frac{2}{3}$ is not present in our simplified form, and the argument of the logarithm is different.

Choice b: $\frac{(\log (x-4))^2}{(\log (x))^3}$ This choice represents a ratio of squared and cubed logarithmic terms, which is structurally different from our simplified expression. Our expression involves a single logarithm of a quotient, not a ratio of logarithms.

Choice c: $\log \left(\frac{(x-4)2}{x3}\right)$ This choice perfectly matches our transformed expression. It has the same logarithmic form and the same argument, confirming its equivalence.

Choice d: None of the above. Since we've found a matching choice, this option is incorrect.

Therefore, after careful analysis, we can confidently conclude that choice c is the equivalent expression to $2 \log (x-4)-3 \log (x)$. This process demonstrates the importance of both transforming the expression and meticulously comparing it to the given options.

In this comprehensive exploration, we've successfully navigated the realm of logarithmic expressions, focusing on transforming and simplifying them using fundamental logarithmic properties. Our journey centered around the expression $2 \log (x-4)-3 \log (x)$, where we aimed to determine its equivalent form from a set of given choices. We began by solidifying our understanding of key logarithmic properties, including the power rule, product rule, and quotient rule. These properties served as the foundation for our transformation process.

We then embarked on a step-by-step transformation of the expression, applying the power rule to move coefficients as exponents and the quotient rule to combine logarithmic terms into a single logarithm of a quotient. This process led us to the simplified form $\log \left(\frac{(x-4)2}{x3}\right)$.

Subsequently, we analyzed the provided choices, meticulously comparing each option to our transformed expression. Through this analysis, we confidently identified choice c, $\log \left(\frac{(x-4)2}{x3}\right)$, as the equivalent expression. This conclusion underscores the importance of both mastering logarithmic properties and employing a systematic approach to problem-solving.

This exploration not only reinforces the practical application of logarithmic properties but also highlights the power of mathematical transformations in revealing underlying relationships and simplifying complex expressions. The ability to manipulate and simplify logarithmic expressions is a valuable asset in various mathematical contexts, including algebra, calculus, and beyond. As you continue your mathematical journey, the skills and insights gained from this exploration will undoubtedly serve you well in tackling a wide range of challenges.