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) = \log_2 4 +

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 the product rule for logarithmic equations and identify which of the given options illustrates this rule.

What is the Product Rule for Logarithmic Equations?

The product rule for logarithmic equations is a fundamental concept that 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:

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

where $b$ is the base of the logarithm, and $x$ and $y$ are the individual factors.

Applying the Product Rule to Logarithmic Equations

To apply the product rule to logarithmic equations, we need to identify the individual factors and then take the logarithm of each factor separately. Once we have the logarithms of the individual factors, we can add them together to get the logarithm of the product.

Analyzing the Options

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

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

This option does not illustrate the product rule for logarithmic equations. The product rule states that the logarithm of a product is equal to the sum of the logarithms of the individual factors, but this option subtracts the logarithm of $x$ from the logarithm of 4.

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

This option also does not illustrate the product rule for logarithmic equations. The product rule states that the logarithm of a product is equal to the sum of the logarithms of the individual factors, but this option multiplies the logarithms of 4 and $x$.

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

This option does not illustrate the product rule for logarithmic equations. As mentioned earlier, the product rule states that the logarithm of a product is equal to the sum of the logarithms of the individual factors, but this option subtracts the logarithm of $x$ from the logarithm of 4.

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

This option illustrates the product rule for logarithmic equations. The product rule states that the logarithm of a product is equal to the sum of the logarithms of the individual factors, and this option correctly adds the logarithms of 4 and $x$ to get the logarithm of $4x$.

Conclusion

Introduction

In our previous article, we explored the product rule for logarithmic equations and identified which of the given options illustrates this rule. In this article, we will answer some frequently asked questions (FAQs) 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:

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

where $b$ is the base of the logarithm, and $x$ and $y$ are the individual factors.

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

A: To apply the product rule to logarithmic equations, you need to identify the individual factors and then take the logarithm of each factor separately. Once you have the logarithms of the individual factors, you can add them together to get the logarithm of the product.

Q: What is the difference between the product rule and the quotient rule for logarithmic equations?

A: The product rule and the quotient rule are two different rules for logarithmic equations. The product rule states that the logarithm of a product is equal to the sum of the logarithms of the individual factors, while the quotient rule states that the logarithm of a quotient is equal to the difference of the logarithms of the individual factors.

Q: Can I use the product rule to simplify logarithmic expressions?

A: Yes, you can use the product rule to simplify logarithmic expressions. By applying the product rule, you can rewrite a logarithmic expression as the sum of the logarithms of the individual factors, which can make it easier to simplify.

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

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

• Subtracting the logarithms of the individual factors instead of adding them
• Multiplying the logarithms of the individual factors instead of adding them
• Failing to identify the individual factors and take the logarithm of each factor separately

Q: How do I know when to use the product rule and when to use the quotient rule?

A: To determine whether to use the product rule or the quotient rule, you need to look at the expression and identify whether it is a product or a quotient. If it is a product, you can use the product rule. If it is a quotient, you can use the quotient rule.

Q: Can I use the product rule with different bases?

A: Yes, you can use the product rule with different bases. The product rule applies to any base, as long as the base is the same for all the logarithms.

Conclusion

In conclusion, the product rule for logarithmic equations is a fundamental concept that states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. By understanding the product rule and how to apply it, you can simplify logarithmic expressions and solve a wide range of mathematical problems.