Choose The Best Answer That Represents The Property Used To Rewrite The Expression. Log ⁡ 2 10 + Log ⁡ 2 4 = Log ⁡ 2 40 \log _2 10 + \log _2 4 = \log _2 40 Lo G 2 ​ 10 + Lo G 2 ​ 4 = Lo G 2 ​ 40 A. Product Property B. Commutative Property C. Power Property D. Quotient Property

Introduction

Logarithmic expressions are a fundamental concept in mathematics, and understanding the properties that govern them is crucial for simplifying and solving equations. In this article, we will delve into the world of logarithmic properties, focusing on the property used to rewrite the expression $\log _2 10 + \log _2 4 = \log _2 40$. We will explore the different properties, including the Product Property, Commutative Property, Power Property, and Quotient Property, to determine which one is applicable in this scenario.

Understanding Logarithmic Properties

Before we dive into the specific property used to rewrite the expression, let's briefly review the four logarithmic properties:

Product Property

The Product Property states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. Mathematically, this can be expressed as:

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

Commutative Property

The Commutative Property states that the order of the factors does not change the result. In other words, the logarithm of a product is the same regardless of the order of the factors. Mathematically, this can be expressed as:

$\log_b x + \log_b y = \log_b y + \log_b x$

Power Property

The Power Property states that the logarithm of a power is equal to the exponent multiplied by the logarithm of the base. Mathematically, this can be expressed as:

$\log_b x^y = y \log_b x$

Quotient Property

The Quotient Property states that the logarithm of a quotient is equal to the logarithm of the dividend minus the logarithm of the divisor. Mathematically, this can be expressed as:

$\log_b \frac{x}{y} = \log_b x - \log_b y$

Applying the Properties to the Expression

Now that we have reviewed the four logarithmic properties, let's apply them to the expression $\log _2 10 + \log _2 4 = \log _2 40$. We can start by examining the left-hand side of the equation:

$\log _2 10 + \log _2 4$

Using the Product Property, we can rewrite this expression as:

$\log _2 (10 \cdot 4)$

Simplifying the expression inside the logarithm, we get:

$\log _2 40$

This is the same as the right-hand side of the equation. Therefore, we can conclude that the Product Property is the property used to rewrite the expression.

Conclusion

In conclusion, the Product Property is the property used to rewrite the expression $\log _2 10 + \log _2 4 = \log _2 40$. This property states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. Understanding the different logarithmic properties is crucial for simplifying and solving equations, and this article has provided a comprehensive review of the four properties.

Choosing the Best Answer

Based on our analysis, the best answer that represents the property used to rewrite the expression is:

A. Product Property

The other options, B. Commutative Property, C. Power Property, and D. Quotient Property, are not applicable in this scenario.

Final Thoughts

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Introduction

In our previous article, we explored the world of logarithmic properties, focusing on the Product Property, Commutative Property, Power Property, and Quotient Property. We applied these properties to the expression $\log _2 10 + \log _2 4 = \log _2 40$ and determined that the Product Property is the property used to rewrite the expression. In this article, we will provide a comprehensive Q&A guide to logarithmic properties, covering a range of topics and questions.

Q&A: Logarithmic Properties

Q: What is the Product Property of logarithms?

A: The Product Property states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. Mathematically, this can be expressed as:

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

Q: What is the Commutative Property of logarithms?

A: The Commutative Property states that the order of the factors does not change the result. In other words, the logarithm of a product is the same regardless of the order of the factors. Mathematically, this can be expressed as:

$\log_b x + \log_b y = \log_b y + \log_b x$

Q: What is the Power Property of logarithms?

A: The Power Property states that the logarithm of a power is equal to the exponent multiplied by the logarithm of the base. Mathematically, this can be expressed as:

$\log_b x^y = y \log_b x$

Q: What is the Quotient Property of logarithms?

A: The Quotient Property states that the logarithm of a quotient is equal to the logarithm of the dividend minus the logarithm of the divisor. Mathematically, this can be expressed as:

$\log_b \frac{x}{y} = \log_b x - \log_b y$

Q: How do I apply the Product Property to simplify an expression?

A: To apply the Product Property, simply multiply the factors inside the logarithm and then take the logarithm of the result. For example:

$\log_2 (10 \cdot 4) = \log_2 40$

Q: Can I use the Commutative Property to simplify an expression?

A: Yes, you can use the Commutative Property to simplify an expression by rearranging the factors. For example:

$\log_2 10 + \log_2 4 = \log_2 4 + \log_2 10$

Q: How do I apply the Power Property to simplify an expression?

A: To apply the Power Property, simply multiply the exponent by the logarithm of the base. For example:

$\log_2 10^3 = 3 \log_2 10$

Q: Can I use the Quotient Property to simplify an expression?

A: Yes, you can use the Quotient Property to simplify an expression by subtracting the logarithm of the divisor from the logarithm of the dividend. For example:

$\log_2 \frac{10}{4} = \log_2 10 - \log_2 4$

Conclusion

In conclusion, logarithmic properties are a fundamental concept in mathematics, and understanding the Product Property, Commutative Property, Power Property, and Quotient Property is crucial for simplifying and solving equations. This Q&A guide has provided a comprehensive review of these properties, and we hope that it has been helpful in understanding the concept of logarithmic properties.

Final Thoughts

Logarithmic expressions are a fundamental concept in mathematics, and understanding the properties that govern them is crucial for simplifying and solving equations. By applying the Product Property, Commutative Property, Power Property, and Quotient Property, we can simplify complex expressions and solve equations with ease. We hope that this Q&A guide has been helpful in understanding the concept of logarithmic properties and has provided a comprehensive review of the four properties.