11.Expand $\log (x / Y)$a) $(\log X) / 3$ B) \$\log X - \log Y$[/tex\] C) $y \log X$ D) $(\log 2) / (\log Y)$

Introduction

Logarithmic expressions are a fundamental concept in mathematics, and expanding them is a crucial skill for students and professionals alike. In this article, we will explore the expansion of the logarithmic expression $\log (x / y)$ and examine the different options provided.

Understanding Logarithmic Expressions

Before we dive into the expansion of $\log (x / y)$, it's essential to understand the basics of logarithmic expressions. A logarithmic expression is a mathematical operation that represents the power to which a base number must be raised to obtain a given value. In other words, if $y = \log_b x$, then $b^y = x$. The most common base used in logarithmic expressions is the base 10, denoted as $\log_{10}$, and the base $e$, denoted as $\ln$.

Expanding $\log (x / y)$

Now that we have a solid understanding of logarithmic expressions, let's focus on expanding $\log (x / y)$. There are several ways to expand this expression, and we will examine each option provided.

a) $(\log x) / 3$

This option is incorrect because it does not accurately represent the expansion of $\log (x / y)$. The correct expansion of $\log (x / y)$ is not a simple division of the logarithm of $x$ by 3.

b) $\log x - \log y$

This option is correct because it accurately represents the expansion of $\log (x / y)$. Using the properties of logarithms, we can rewrite $\log (x / y)$ as $\log x - \log y$. This is because the logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator.

c) $y \log x$

This option is incorrect because it does not accurately represent the expansion of $\log (x / y)$. The correct expansion of $\log (x / y)$ is not a simple multiplication of $y$ and the logarithm of $x$.

d) $(\log 2) / (\log y)$

This option is incorrect because it does not accurately represent the expansion of $\log (x / y)$. The correct expansion of $\log (x / y)$ is not a simple division of the logarithm of 2 by the logarithm of $y$.

Conclusion

In conclusion, the correct expansion of $\log (x / y)$ is $\log x - \log y$. This is because the logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator. Understanding the properties of logarithms is essential for expanding logarithmic expressions, and this article has provided a comprehensive guide to help students and professionals alike.

Properties of Logarithms

The properties of logarithms are essential for expanding logarithmic expressions. Here are some of the key properties:

• Product Property: $\log (xy) = \log x + \log y$
• Quotient Property: $\log (x / y) = \log x - \log y$
• Power Property: $\log (x^y) = y \log x$
• Change of Base Property: $\log_b x = \frac{\log_c x}{\log_c b}$

Real-World Applications

Logarithmic expressions have numerous real-world applications. Here are a few examples:

• Finance: Logarithmic expressions are used to calculate interest rates and investment returns.
• Science: Logarithmic expressions are used to calculate pH levels and concentrations of solutions.
• Engineering: Logarithmic expressions are used to calculate sound levels and signal strengths.

Practice Problems

Here are some practice problems to help you reinforce your understanding of logarithmic expressions:

• Problem 1: Expand $\log (x^2 / y^3)$ using the properties of logarithms.
• Problem 2: Simplify $\log (2^3 \cdot 3^2)$ using the properties of logarithms.
• Problem 3: Calculate the value of $\log (1000)$ using the properties of logarithms.

Conclusion

Q&A: Expanding Logarithmic Expressions

Q: What is the correct expansion of $\log (x / y)$?

A: The correct expansion of $\log (x / y)$ is $\log x - \log y$. This is because the logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator.

Q: What is the product property of logarithms?

A: The product property of logarithms states that $\log (xy) = \log x + \log y$. This means that the logarithm of a product is equal to the sum of the logarithms of the individual factors.

Q: What is the power property of logarithms?

A: The power property of logarithms states that $\log (x^y) = y \log x$. This means that the logarithm of a power is equal to the exponent multiplied by the logarithm of the base.

Q: How do I simplify $\log (2^3 \cdot 3^2)$ using the properties of logarithms?

A: To simplify $\log (2^3 \cdot 3^2)$, we can use the product property of logarithms, which states that $\log (xy) = \log x + \log y$. Therefore, we can rewrite $\log (2^3 \cdot 3^2)$ as $\log 2^3 + \log 3^2$. Using the power property of logarithms, we can simplify this expression further to $3 \log 2 + 2 \log 3$.

Q: How do I calculate the value of $\log (1000)$ using the properties of logarithms?

A: To calculate the value of $\log (1000)$, we can use the fact that $1000 = 10^3$. Therefore, we can rewrite $\log (1000)$ as $\log 10^3$. Using the power property of logarithms, we can simplify this expression to $3 \log 10$. Since $\log 10 = 1$, we can conclude that $\log (1000) = 3$.

Q: What is the change of base property of logarithms?

A: The change of base property of logarithms states that $\log_b x = \frac{\log_c x}{\log_c b}$. This means that we can change the base of a logarithm by dividing the logarithm of the number by the logarithm of the base.

Q: How do I use the change of base property to simplify $\log_2 8$?

A: To simplify $\log_2 8$, we can use the change of base property, which states that $\log_b x = \frac{\log_c x}{\log_c b}$. Therefore, we can rewrite $\log_2 8$ as $\frac{\log 8}{\log 2}$. Using the fact that $8 = 2^3$, we can rewrite $\log 8$ as $\log 2^3$. Using the power property of logarithms, we can simplify this expression to $3 \log 2$. Therefore, we can conclude that $\log_2 8 = \frac{3 \log 2}{\log 2} = 3$.

Q: What are some real-world applications of logarithmic expressions?

A: Logarithmic expressions have numerous real-world applications, including:

• Finance: Logarithmic expressions are used to calculate interest rates and investment returns.
• Science: Logarithmic expressions are used to calculate pH levels and concentrations of solutions.
• Engineering: Logarithmic expressions are used to calculate sound levels and signal strengths.

Conclusion

In conclusion, logarithmic expressions are a fundamental concept in mathematics, and expanding them is a crucial skill for students and professionals alike. This article has provided a comprehensive guide to help you understand the properties of logarithms and expand logarithmic expressions. With practice and patience, you will become proficient in expanding logarithmic expressions and applying them to real-world problems.