Expand Log Base 7 Of (x^3 / Y)^3: A Step-by-Step Guide

In the realm of mathematics, logarithms serve as powerful tools for simplifying complex expressions and solving equations. Mastering the art of expanding logarithmic expressions is a crucial skill for anyone venturing into advanced mathematical concepts. In this comprehensive guide, we'll delve into the intricacies of expanding logarithmic expressions, using the expression $\log _7\left(x^3 / y\right)^3$ as our illustrative example. Our goal is to break down the process step-by-step, ensuring a clear understanding of the underlying principles and techniques involved. This will not only help in solving this particular problem but also provide a strong foundation for tackling more complex logarithmic problems in the future. By understanding the properties of logarithms and how to apply them correctly, you can transform seemingly complicated expressions into simpler, more manageable forms. This skill is invaluable in various fields, including calculus, physics, engineering, and computer science, where logarithmic scales and transformations are frequently used to model and analyze data.

Understanding the Fundamentals of Logarithms

Before we dive into the expansion of the given logarithmic expression, let's take a moment to solidify our understanding of the fundamental principles of logarithms. Logarithms are essentially the inverse operation 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. Mathematically, this relationship is expressed as follows:

If $b^y = x$, then $\log_b x = y$

Here,

• b represents the base of the logarithm.
• x is the argument of the logarithm (the number whose logarithm is being taken).
• y is the exponent or the logarithm itself.

To truly grasp the concept, let's consider a concrete example. Suppose we have $2^3 = 8$. In logarithmic form, this can be written as $\log_2 8 = 3$. This tells us that the logarithm of 8 to the base 2 is 3, meaning that 2 must be raised to the power of 3 to obtain 8. This fundamental understanding of the inverse relationship between exponentiation and logarithms is the bedrock upon which all logarithmic manipulations are built. Without a solid grasp of this concept, expanding and simplifying logarithmic expressions can become a daunting task. It's also crucial to remember that the base of a logarithm must be a positive number not equal to 1, and the argument of the logarithm must be a positive number. These restrictions are in place to ensure that the logarithmic function is well-defined and behaves predictably.

Key Properties of Logarithms for Expansion

To effectively expand logarithmic expressions, we must be well-versed in the fundamental properties that govern their behavior. These properties act as the rules of the game, allowing us to manipulate logarithmic expressions in a mathematically sound manner. Let's explore the key properties that are most relevant to our task of expanding $\log _7\left(x^3 / y\right)^3$:

1. Power Rule: This property is perhaps the most crucial for expanding expressions involving exponents within logarithms. The power 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:

   $\log_b(x^p) = p \log_b(x)$

   This rule allows us to bring exponents outside the logarithm, which is a key step in expansion. For instance, $\log_2(4^3)$ can be rewritten as $3 \log_2(4)$.

2. Product Rule: This property deals with the logarithm of a product. It states that the logarithm of the product of two numbers is equal to the sum of the logarithms of the individual numbers. Mathematically:

   $\log_b(xy) = \log_b(x) + \log_b(y)$

   This rule enables us to separate the logarithm of a product into a sum of logarithms, further expanding the expression. For example, $\log_3(9 \cdot 27)$ can be rewritten as $\log_3(9) + \log_3(27)$.

3. Quotient Rule: This property is the counterpart to the product rule and deals with the logarithm of a quotient. It states that the logarithm of the quotient of two numbers is equal to the difference of the logarithms of the individual numbers. Mathematically:

   $\log_b(x/y) = \log_b(x) - \log_b(y)$

   This rule allows us to separate the logarithm of a division into a difference of logarithms, which is essential for expanding expressions with fractions. For example, $\log_5(25/5)$ can be rewritten as $\log_5(25) - \log_5(5)$.

These three properties, the power rule, the product rule, and the quotient rule, are the cornerstones of expanding logarithmic expressions. Mastering these rules and understanding how to apply them in various scenarios is paramount to success in logarithmic manipulations. They allow us to break down complex logarithmic expressions into simpler, more manageable components, making them easier to analyze and solve.

Step-by-Step Expansion of $\log _7\left(x^3 / y\right)^3$

Now that we have a firm grasp of the fundamental properties of logarithms, let's apply them to expand the given expression, $\log _7\left(x^3 / y\right)^3$. We'll break down the process into a series of clear, concise steps, ensuring that each step is justified by the properties we've discussed. This methodical approach will not only lead us to the correct answer but also reinforce our understanding of how to effectively manipulate logarithmic expressions.

Step 1: Applying the Power Rule

The first step in expanding this expression is to utilize the power rule. We have the entire term $\left(x^3 / y\right)$ raised to the power of 3 within the logarithm. According to the power rule, we can bring this exponent outside the logarithm as a coefficient:

$\log _7\left(x^3 / y\right)^3 = 3 \log _7\left(x^3 / y\right)$

This step significantly simplifies the expression by removing the outer exponent, making it easier to apply the subsequent properties. It's a crucial first step in untangling the logarithmic expression and setting the stage for further expansion.

Step 2: Applying the Quotient Rule

Next, we encounter a quotient within the logarithm: $x^3 / y$. The quotient rule allows us to separate the logarithm of a quotient into the difference of the logarithms of the numerator and the denominator. Applying the quotient rule, we get:

$3 \log _7\left(x^3 / y\right) = 3 \left[\log _7\left(x^3\right) - \log _7\left(y\right)\right]$

Notice that we've kept the coefficient 3 outside the brackets, as it applies to the entire expression within the logarithm. This step further simplifies the expression by separating the numerator and denominator, allowing us to deal with them individually.

Step 3: Applying the Power Rule Again

We now have $\log _7\left(x^3\right)$ within the expression. Once again, we can apply the power rule to bring the exponent 3 outside the logarithm:

$3 \left[\log _7\left(x^3\right) - \log _7\left(y\right)\right] = 3 \left[3 \log _7\left(x\right) - \log _7\left(y\right)\right]$

This step isolates the variable x within the logarithm, making the expression even simpler.

Step 4: Distributing the Coefficient

Finally, we need to distribute the coefficient 3 that's outside the brackets to both terms inside the brackets:

$3 \left[3 \log _7\left(x\right) - \log _7\left(y\right)\right] = 9 \log _7 x - 3 \log _7 y$

This final step completes the expansion process, leaving us with a simplified expression where all the logarithmic terms are separated.

Therefore, the expanded form of $\log _7\left(x^3 / y\right)^3$ is $9 \log _7 x - 3 \log _7 y$.

Analyzing the Answer Choices

Now that we've meticulously expanded the logarithmic expression, let's compare our result with the given answer choices to identify the correct one. Our expanded expression is:

$9 \log _7 x - 3 \log _7 y$

Let's examine the answer choices:

A. $9 \log _7 x - 3 \log _7 y$ B. $3 \log _7 x - 3 \log _7 y$ C. $9 \log _7 x - \log _7 y$ D. $9 \log _7 x + 3 \log _7 y$

By direct comparison, we can clearly see that answer choice A, $9 \log _7 x - 3 \log _7 y$, perfectly matches our expanded expression. Therefore, answer choice A is the correct answer.

The other answer choices are incorrect due to errors in applying the logarithmic properties or in the distribution of coefficients. For instance, answer choice B has an incorrect coefficient for the first term, while answer choice C has an incorrect coefficient for the second term. Answer choice D has the correct coefficients but an incorrect sign between the terms, indicating a misunderstanding of the quotient rule.

This step-by-step analysis not only helps us identify the correct answer but also reinforces our understanding of the expansion process and the importance of accurately applying the logarithmic properties.

Common Mistakes to Avoid When Expanding Logarithms

Expanding logarithms can sometimes be tricky, and it's easy to fall into common traps if you're not careful. To ensure accuracy and avoid errors, let's highlight some of the most frequent mistakes that students make when expanding logarithmic expressions. Being aware of these pitfalls will help you approach logarithmic problems with greater confidence and precision.

1. Incorrectly Applying the Power Rule: A common mistake is to apply the power rule to terms that are not actually raised to a power within the logarithm. Remember, the power rule applies only when the entire argument of the logarithm is raised to a power. For example, $\log_b(x + y)^2$ can be expanded using the power rule, but $\log_b(x^2 + y^2)$ cannot. In the latter case, the power rule cannot be directly applied because the entire argument is not raised to the power of 2.
2. Misunderstanding the Product and Quotient Rules: Students often confuse the product and quotient rules, leading to incorrect expansions. Remember that the product rule applies to the logarithm of a product, resulting in a sum of logarithms, while the quotient rule applies to the logarithm of a quotient, resulting in a difference of logarithms. Mixing these rules up can lead to significant errors in the expansion process. For example, incorrectly applying the product rule to a quotient would result in an incorrect expansion.
3. Forgetting to Distribute Coefficients: When a coefficient is present outside a logarithmic expression, it must be distributed to all terms within the expression after expansion. Failing to distribute the coefficient can lead to an incomplete or incorrect answer. For example, if you have $2[\log_b(x) + \log_b(y)]$, you must distribute the 2 to both terms, resulting in $2\log_b(x) + 2\log_b(y)$.
4. Applying Rules in the Wrong Order: The order in which you apply the logarithmic properties can sometimes affect the complexity of the expansion process. It's generally best to apply the power rule first, followed by the product or quotient rule. Applying the rules in the wrong order can sometimes lead to more complicated expressions that are harder to simplify. However, the final answer should be the same regardless of the order in which the rules are applied, as long as they are applied correctly.
5. Ignoring the Base of the Logarithm: The base of the logarithm is a crucial part of the expression and should not be ignored. Always ensure that you are working with the correct base and that your manipulations are consistent with that base. For example, $\log_2(8)$ is different from $\log_{10}(8)$, and you cannot directly compare or combine logarithms with different bases without converting them to a common base.

By being mindful of these common mistakes and practicing the correct application of logarithmic properties, you can significantly improve your accuracy and proficiency in expanding logarithmic expressions.

Conclusion: Mastering Logarithmic Expansion

In conclusion, expanding logarithmic expressions is a fundamental skill in mathematics with wide-ranging applications. By understanding the core properties of logarithms – the power rule, the product rule, and the quotient rule – and applying them systematically, we can transform complex logarithmic expressions into simpler, more manageable forms. Our step-by-step expansion of $\log _7\left(x^3 / y\right)^3$ serves as a practical illustration of this process, highlighting the importance of each property and the order in which they are applied. The correct answer, $9 \log _7 x - 3 \log _7 y$, demonstrates the power of these rules in simplifying logarithmic expressions.

Moreover, we've discussed common mistakes to avoid when expanding logarithms, such as incorrectly applying the power rule, misunderstanding the product and quotient rules, forgetting to distribute coefficients, applying rules in the wrong order, and ignoring the base of the logarithm. Being aware of these pitfalls is crucial for maintaining accuracy and avoiding errors in your calculations.

Mastering logarithmic expansion not only enhances your mathematical abilities but also equips you with a valuable tool for solving problems in various fields, including science, engineering, and finance. The ability to manipulate logarithmic expressions effectively is essential for working with logarithmic scales, solving exponential equations, and analyzing data that exhibits exponential growth or decay.

Therefore, continue to practice expanding logarithmic expressions, paying close attention to the properties and potential pitfalls. With consistent effort and a solid understanding of the underlying principles, you'll become proficient in this essential mathematical skill and unlock its numerous applications.