Which Expressions Are Equivalent To The One Below? Check All That Apply.$\log 2 - \log 8$A. $\log \left(\frac{1}{4}\right$\] B. $\log 2$ C. $\log 4$ D. $\log (2) + \log \left(\frac{1}{8}\right$\]

**Understanding Logarithmic Expressions: A Comprehensive Guide** ===========================================================

Introduction

Logarithmic expressions are a fundamental concept in mathematics, and understanding them is crucial for solving various mathematical problems. In this article, we will explore the concept of logarithmic expressions, their properties, and how to simplify them. We will also examine the given expression $\log 2 - \log 8$ and determine which of the provided options are equivalent to it.

What are Logarithmic Expressions?

A logarithmic expression is a mathematical expression that represents the power to which a base number must be raised to obtain a given value. In other words, it is the inverse operation of exponentiation. For example, $\log_2 8$ represents the power to which 2 must be raised to obtain 8.

Properties of Logarithmic Expressions

There are several properties of logarithmic expressions that are essential to understand:

• Product Property: $\log_b (xy) = \log_b x + \log_b y$
• Quotient Property: $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$
• Power Property: $\log_b x^y = y \log_b x$

Simplifying Logarithmic Expressions

To simplify a logarithmic expression, we can use the properties mentioned above. For example, to simplify $\log 2 - \log 8$, we can use the quotient property:

$\log 2 - \log 8 = \log \left(\frac{2}{8}\right) = \log \left(\frac{1}{4}\right)$

Which Expressions are Equivalent to $\log 2 - \log 8$?

Now that we have simplified the expression $\log 2 - \log 8$, let's examine the provided options and determine which ones are equivalent to it.

A. $\log \left(\frac{1}{4}\right)$

This option is equivalent to the simplified expression $\log \left(\frac{1}{4}\right)$.

B. $\log 2$

This option is not equivalent to the simplified expression $\log \left(\frac{1}{4}\right)$.

C. $\log 4$

This option is not equivalent to the simplified expression $\log \left(\frac{1}{4}\right)$.

D. $\log (2) + \log \left(\frac{1}{8}\right)$

This option is not equivalent to the simplified expression $\log \left(\frac{1}{4}\right)$.

Conclusion

In conclusion, the only option that is equivalent to the expression $\log 2 - \log 8$ is $\log \left(\frac{1}{4}\right)$. Understanding logarithmic expressions and their properties is crucial for solving various mathematical problems. By applying the properties of logarithmic expressions, we can simplify complex expressions and determine which ones are equivalent to a given expression.

Frequently Asked Questions

Q: What is the difference between a logarithmic expression and an exponential expression?

A: A logarithmic expression represents the power to which a base number must be raised to obtain a given value, while an exponential expression represents the result of raising a base number to a given power.

Q: How do I simplify a logarithmic expression?

A: To simplify a logarithmic expression, you can use the properties of logarithmic expressions, such as the product property, quotient property, and power property.

Q: What is the quotient property of logarithmic expressions?

A: The quotient property of logarithmic expressions states that $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$.

Q: How do I determine which expressions are equivalent to a given logarithmic expression?

A: To determine which expressions are equivalent to a given logarithmic expression, you can simplify the expression using the properties of logarithmic expressions and compare it to the given expression.

Q: What is the product property of logarithmic expressions?

A: The product property of logarithmic expressions states that $\log_b (xy) = \log_b x + \log_b y$.

Q: How do I use the power property of logarithmic expressions?

A: To use the power property of logarithmic expressions, you can multiply the exponent by the logarithm of the base, such as $\log_b x^y = y \log_b x$.