Which Of The Following Illustrates The Product Rule For Logarithmic Equations?A. Log ⁡ 2 ( 4 X ) = Log ⁡ 2 4 + Log ⁡ 2 X \log_2(4x) = \log_2 4 + \log_2 X Lo G 2 ​ ( 4 X ) = Lo G 2 ​ 4 + Lo G 2 ​ X B. Log ⁡ 2 ( 4 X ) = Log ⁡ 2 4 ⋅ Log ⁡ 2 X \log_2(4x) = \log_2 4 \cdot \log_2 X Lo G 2 ​ ( 4 X ) = Lo G 2 ​ 4 ⋅ Lo G 2 ​ X C. Log ⁡ 2 ( 4 X ) = Log ⁡ 2 4 − Log ⁡ 2 X \log_2(4x) = \log_2 4 - \log_2 X Lo G 2 ​ ( 4 X ) = Lo G 2 ​ 4 − Lo G 2 ​ X D. $\log_2(4x) =

Introduction

Logarithmic equations are a fundamental concept in mathematics, and understanding the rules that govern them is crucial for solving various mathematical problems. One of the essential rules in logarithmic equations is the product rule, which states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. In this article, we will explore which of the given options illustrates the product rule for logarithmic equations.

What is the Product Rule for Logarithmic Equations?

The product rule for logarithmic equations is a fundamental concept that helps us simplify complex logarithmic expressions. It 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:

$$\log_b (m \cdot n) = \log_b m + \log_b n$$

where $b$ is the base of the logarithm, and $m$ and $n$ are the individual factors.

Analyzing the Options

Now that we have a clear understanding of the product rule for logarithmic equations, let's analyze the given options to determine which one illustrates this rule.

Option A: $\log_2(4x) = \log_2 4 + \log_2 x$

This option suggests that the logarithm of $4x$ is equal to the sum of the logarithms of $4$ and $x$. However, this is not a correct application of the product rule, as the product rule states that the logarithm of a product is equal to the sum of the logarithms of the individual factors, not the sum of the logarithms of the individual factors and a constant.

Option B: $\log_2(4x) = \log_2 4 \cdot \log_2 x$

This option suggests that the logarithm of $4x$ is equal to the product of the logarithms of $4$ and $x$. However, this is not a correct application of the product rule, as the product rule states that the logarithm of a product is equal to the sum of the logarithms of the individual factors, not the product of the logarithms of the individual factors.

Option C: $\log_2(4x) = \log_2 4 - \log_2 x$

This option suggests that the logarithm of $4x$ is equal to the difference of the logarithms of $4$ and $x$. However, this is not a correct application of the product rule, as the product rule states that the logarithm of a product is equal to the sum of the logarithms of the individual factors, not the difference of the logarithms of the individual factors.

Option D: $\log_2(4x) = \log_2 4 + \log_2 x$

This option suggests that the logarithm of $4x$ is equal to the sum of the logarithms of $4$ and $x$. This is a correct application of the product rule, as the product rule states that the logarithm of a product is equal to the sum of the logarithms of the individual factors.

Conclusion

In conclusion, the correct option that illustrates the product rule for logarithmic equations is Option D: $\log_2(4x) = \log_2 4 + \log_2 x$. This option correctly applies the product rule, which states that the logarithm of a product is equal to the sum of the logarithms of the individual factors.

Example Problems

To further illustrate the product rule for logarithmic equations, let's consider a few example problems.

Example Problem 1

Simplify the expression: $\log_2 (6x)$

Using the product rule, we can rewrite the expression as:

$$\log_2 (6x) = \log_2 6 + \log_2 x$$

Example Problem 2

Simplify the expression: $\log_2 (9y)$

Using the product rule, we can rewrite the expression as:

$$\log_2 (9y) = \log_2 9 + \log_2 y$$

Tips and Tricks

When working with logarithmic equations, it's essential to remember the product rule, which states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. By applying this rule, you can simplify complex logarithmic expressions and solve various mathematical problems.

Common Mistakes

When working with logarithmic equations, it's easy to make mistakes. Here are a few common mistakes to avoid:

• Forgetting to apply the product rule: Make sure to apply the product rule when simplifying logarithmic expressions.
• Incorrectly applying the product rule: Double-check your work to ensure that you're applying the product rule correctly.
• Not simplifying the expression: Make sure to simplify the expression as much as possible to avoid unnecessary complexity.

Conclusion

Introduction

In our previous article, we explored the product rule for logarithmic equations, which states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. In this article, we will answer some frequently asked questions about the product rule for logarithmic equations.

Q: What is the product rule for logarithmic equations?

A: The product rule for logarithmic equations 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:

$$\log_b (m \cdot n) = \log_b m + \log_b n$$

Q: How do I apply the product rule for logarithmic equations?

A: To apply the product rule for logarithmic equations, simply identify the product and the individual factors, and then rewrite the expression as the sum of the logarithms of the individual factors.

Q: What are some common mistakes to avoid when applying the product rule for logarithmic equations?

A: Some common mistakes to avoid when applying the product rule for logarithmic equations include:

• Forgetting to apply the product rule: Make sure to apply the product rule when simplifying logarithmic expressions.
• Incorrectly applying the product rule: Double-check your work to ensure that you're applying the product rule correctly.
• Not simplifying the expression: Make sure to simplify the expression as much as possible to avoid unnecessary complexity.

Q: Can I use the product rule for logarithmic equations with any base?

A: Yes, the product rule for logarithmic equations can be used with any base. The base of the logarithm is not relevant to the product rule.

Q: How do I simplify logarithmic expressions using the product rule?

A: To simplify logarithmic expressions using the product rule, simply identify the product and the individual factors, and then rewrite the expression as the sum of the logarithms of the individual factors.

Q: Can I use the product rule for logarithmic equations with logarithms of different bases?

A: Yes, the product rule for logarithmic equations can be used with logarithms of different bases. The product rule states that the logarithm of a product is equal to the sum of the logarithms of the individual factors, regardless of the base.

Q: What are some real-world applications of the product rule for logarithmic equations?

A: The product rule for logarithmic equations has many real-world applications, including:

• Engineering: The product rule for logarithmic equations is used in engineering to simplify complex logarithmic expressions and solve problems involving logarithmic functions.
• Computer Science: The product rule for logarithmic equations is used in computer science to simplify complex logarithmic expressions and solve problems involving logarithmic functions.
• Finance: The product rule for logarithmic equations is used in finance to simplify complex logarithmic expressions and solve problems involving logarithmic functions.

Conclusion

In conclusion, the product rule for logarithmic equations is a fundamental concept that helps us simplify complex logarithmic expressions. By understanding and applying this rule, you can solve various mathematical problems and simplify complex expressions. Remember to always apply the product rule when working with logarithmic equations, and avoid common mistakes such as forgetting to apply the product rule or incorrectly applying it.

Additional Resources

For more information on the product rule for logarithmic equations, check out the following resources:

• Mathway: Mathway is an online math problem solver that can help you with logarithmic equations and other math problems.
• Khan Academy: Khan Academy is a free online resource that offers video lessons and practice exercises on logarithmic equations and other math topics.
• Wolfram Alpha: Wolfram Alpha is a powerful online calculator that can help you with logarithmic equations and other math problems.

Practice Problems

To practice applying the product rule for logarithmic equations, try the following problems:

• Problem 1: Simplify the expression: $\log_2 (6x)$
• Problem 2: Simplify the expression: $\log_2 (9y)$
• Problem 3: Simplify the expression: $\log_3 (12x)$

Answer Key

• Problem 1: $\log_2 (6x) = \log_2 6 + \log_2 x$
• Problem 2: $\log_2 (9y) = \log_2 9 + \log_2 y$
• Problem 3: $\log_3 (12x) = \log_3 12 + \log_3 x$