Rewriting 2 Log_6 10-2 Log_6 7 Using Logarithmic Properties

Introduction

In this article, we will explore how to rewrite the expression $2 \log _6 10 - 2 \log _6 7$ using the properties of logarithms. Logarithms are a fundamental concept in mathematics, and understanding their properties is crucial for simplifying and solving various mathematical problems. This article aims to provide a comprehensive explanation of the steps involved in rewriting the given expression, making it easier to understand and apply logarithmic properties effectively. We will delve into the power rule and quotient rule of logarithms, demonstrating how they are used to simplify the expression into a more concise form. By the end of this article, you will have a clear understanding of how to manipulate logarithmic expressions and apply the relevant properties to achieve simplification. Let’s dive into the world of logarithms and discover how we can rewrite this expression using the powerful tools at our disposal.

Understanding Logarithms

Before diving into the specific problem, let's briefly review the concept of logarithms. A logarithm is the inverse operation to exponentiation. In simpler terms, if we have an equation $b^y = x$, the logarithm of $x$ to the base $b$ is $y$, written as $\log_b x = y$. Understanding this fundamental relationship is crucial for working with logarithmic expressions. Logarithms possess several properties that allow us to manipulate and simplify complex expressions. These properties include the product rule, the quotient rule, and the power rule. Each of these rules provides a specific way to combine or separate logarithmic terms, making it easier to simplify expressions. For instance, the product rule states that the logarithm of a product is the sum of the logarithms, while the quotient rule states that the logarithm of a quotient is the difference of the logarithms. The power rule, which we will use extensively in this article, allows us to move exponents from the argument of the logarithm to the front as a coefficient. Mastering these properties is essential for efficiently working with logarithms and solving related problems. In the following sections, we will apply these properties to rewrite the given logarithmic expression.

Properties of Logarithms

To rewrite the expression $2 \log _6 10 - 2 \log _6 7$, we need to use the properties of logarithms. The two main properties we will use are the power rule and the quotient rule. The power rule states that $a \log_b x = \log_b (x^a)$. This rule allows us to move the coefficient of a logarithm to the exponent of its argument. In our expression, we have $2 \log _6 10$ and $2 \log _6 7$, and we can apply the power rule to both terms. This will transform the expression into $\log _6 (10^2) - \log _6 (7^2)$. Next, we will use the quotient rule, which states that $\log_b x - \log_b y = \log_b \frac{x}{y}$. This rule allows us to combine two logarithms with the same base that are being subtracted into a single logarithm of a quotient. Applying the quotient rule to our transformed expression, $\log _6 (10^2) - \log _6 (7^2)$, we get $\log _6 \frac{10^2}{7^2}$. By understanding and applying these properties, we can effectively simplify and rewrite logarithmic expressions, making them easier to work with and solve. In the next section, we will apply these rules step-by-step to our given expression to arrive at the simplified form.

Applying the Power Rule

The first step in rewriting the expression $2 \log _6 10 - 2 \log _6 7$ is to apply the power rule of logarithms. The power rule states that $a \log_b x = \log_b (x^a)$. This rule allows us to move the coefficient in front of the logarithm to the exponent of the argument inside the logarithm. In our expression, we have two terms: $2 \log _6 10$ and $2 \log _6 7$. Applying the power rule to the first term, $2 \log _6 10$, we get $\log _6 (10^2)$. This means we are taking the coefficient 2 and making it the exponent of 10. Similarly, applying the power rule to the second term, $2 \log _6 7$, we get $\log _6 (7^2)$. Now our expression looks like this: $\log _6 (10^2) - \log _6 (7^2)$. This transformation is a crucial step in simplifying the expression because it allows us to combine the two logarithmic terms into a single logarithm using another property, the quotient rule. By correctly applying the power rule, we have successfully rewritten the expression in a form that is easier to manipulate and simplify further. In the next section, we will apply the quotient rule to combine these logarithmic terms into a single logarithm.

Applying the Quotient Rule

After applying the power rule, our expression is now $\log _6 (10^2) - \log _6 (7^2)$. The next step in simplifying this expression is to apply the quotient rule of logarithms. The quotient rule states that $\log_b x - \log_b y = \log_b \frac{x}{y}$. This rule allows us to combine two logarithms with the same base that are being subtracted into a single logarithm of a quotient. In our expression, we have $\log _6 (10^2)$ and $\log _6 (7^2)$, which are being subtracted, and they both have the same base of 6. Applying the quotient rule, we combine these two logarithms into a single logarithm by dividing the arguments: $\log _6 \frac{10^2}{7^2}$. This transformation significantly simplifies the expression by reducing two logarithmic terms into one. Now, we have a single logarithm, which is much easier to work with. The quotient rule is a powerful tool in simplifying logarithmic expressions, especially when dealing with subtraction of logarithms. By applying the quotient rule correctly, we have successfully rewritten the expression into a more concise and manageable form. In the following sections, we will finalize the simplification and compare our result with the given options.

Final Simplified Expression

After applying both the power rule and the quotient rule, we have successfully rewritten the expression $2 \log _6 10 - 2 \log _6 7$ into a simplified form. Starting with the original expression, we first applied the power rule, which allowed us to rewrite the expression as $\log _6 (10^2) - \log _6 (7^2)$. Next, we applied the quotient rule, which enabled us to combine the two logarithmic terms into a single logarithm, resulting in $\log _6 \frac{10^2}{7^2}$. This is our final simplified expression. It is a single logarithmic term that represents the original expression in a more concise and manageable form. This process demonstrates the power of logarithmic properties in simplifying complex expressions. By understanding and correctly applying these properties, we can transform expressions into forms that are easier to evaluate and manipulate. In the next section, we will compare this simplified expression with the given options to determine the correct answer.

Comparing with the Options

Now that we have rewritten the expression $2 \log _6 10 - 2 \log _6 7$ in its simplified form, $\log _6 \frac{10^2}{7^2}$, we need to compare this result with the given options to identify the correct answer. The options provided are:

A. $\log _6 \frac{10^2}{7^2}$ B. $\log _6 \frac{10^2}{7}$ C. $\log _6 770$ D. $\log _6(10 \cdot 7^2)$

Comparing our simplified expression with the options, we can see that option A, $\log _6 \frac{10^2}{7^2}$, exactly matches our result. This confirms that we have correctly applied the properties of logarithms to rewrite the original expression. The other options do not match our simplified form. Option B, $\log _6 \frac{10^2}{7}$, is incorrect because it does not have $7^2$ in the denominator. Option C, $\log _6 770$, and option D, $\log _6(10 \cdot 7^2)$, are also incorrect as they do not represent the simplified form of the original expression. Therefore, the correct answer is option A. This comparison step is crucial in ensuring that we have not only simplified the expression correctly but also that we have identified the correct answer from the given choices. In the final section, we will summarize the steps and provide a concluding remark.

Conclusion

In this article, we successfully rewrote the expression $2 \log _6 10 - 2 \log _6 7$ using the properties of logarithms. We began by understanding the basic concept of logarithms and their properties, including the power rule and the quotient rule. The power rule allowed us to move the coefficients of the logarithmic terms to the exponents of their arguments, transforming the expression into $\log _6 (10^2) - \log _6 (7^2)$. Then, we applied the quotient rule, which enabled us to combine the two logarithmic terms into a single logarithm, resulting in the simplified expression $\log _6 \frac{10^2}{7^2}$. Finally, we compared our simplified expression with the given options and confirmed that option A, $\log _6 \frac{10^2}{7^2}$, is the correct answer. This exercise demonstrates the importance of understanding and applying the properties of logarithms to simplify complex expressions. By mastering these properties, you can efficiently solve various mathematical problems involving logarithms. This comprehensive approach ensures that the expression is rewritten correctly and matches one of the provided options. Logarithmic properties are powerful tools in mathematics, and their correct application can greatly simplify problem-solving.