Use The Power Property To Rewrite Log ⁡ 3 X 9 \log _3 X^9 Lo G 3 ​ X 9 .A. 3 Log ⁡ 9 X 3 \log _9 X 3 Lo G 9 ​ X B. 9 Log ⁡ 3 X 9 \log _3 X 9 Lo G 3 ​ X C. 3 Log ⁡ X 3 \log X 3 Lo G X D. 2 + Log ⁡ 3 X 2 + \log _3 X 2 + Lo G 3 ​ X

Introduction

Logarithmic expressions are a fundamental concept in mathematics, and understanding how to rewrite them is crucial for solving various mathematical problems. In this article, we will focus on using the power property to rewrite the logarithmic expression $\log _3 x^9$. This property states that if $a$ is a positive real number, then $\log _a x^b = b \log _a x$. We will explore how to apply this property to rewrite the given expression and provide a step-by-step guide on how to do it.

Understanding the Power Property

The power property is a fundamental concept in logarithmic expressions, and it states that if $a$ is a positive real number, then $\log _a x^b = b \log _a x$. This means that when we have a logarithmic expression with a base raised to a power, we can rewrite it as the power multiplied by the logarithm of the base. For example, if we have $\log _2 x^4$, we can rewrite it as $4 \log _2 x$ using the power property.

Applying the Power Property to $\log _3 x^9$

Now that we have a good understanding of the power property, let's apply it to the given expression $\log _3 x^9$. Using the power property, we can rewrite this expression as $9 \log _3 x$. This is because the power property states that $\log _a x^b = b \log _a x$, and in this case, $b = 9$.

Why is the Correct Answer $9 \log _3 x$?

The correct answer is $9 \log _3 x$ because the power property states that $\log _a x^b = b \log _a x$. In this case, $a = 3$, $b = 9$, and $x$ is the variable. Therefore, we can rewrite the expression $\log _3 x^9$ as $9 \log _3 x$ using the power property.

Why are the Other Options Incorrect?

Let's take a closer look at the other options and see why they are incorrect.

• Option A: $3 \log _9 x$. This option is incorrect because the power property states that $\log _a x^b = b \log _a x$, not $\log _a x^b = \log _a x^b$. Therefore, we cannot rewrite the expression $\log _3 x^9$ as $3 \log _9 x$.
• Option C: $3 \log x$. This option is incorrect because the power property states that $\log _a x^b = b \log _a x$, not $\log _a x^b = \log _a x$. Therefore, we cannot rewrite the expression $\log _3 x^9$ as $3 \log x$.
• Option D: $2 + \log _3 x$. This option is incorrect because the power property states that $\log _a x^b = b \log _a x$, not $\log _a x^b = 2 + \log _a x$. Therefore, we cannot rewrite the expression $\log _3 x^9$ as $2 + \log _3 x$.

Conclusion

In conclusion, the power property is a fundamental concept in logarithmic expressions, and understanding how to apply it is crucial for solving various mathematical problems. In this article, we have explored how to use the power property to rewrite the logarithmic expression $\log _3 x^9$. We have also discussed why the other options are incorrect and provided a step-by-step guide on how to apply the power property. By following this guide, you should be able to rewrite logarithmic expressions with ease and solve various mathematical problems.

Frequently Asked Questions

• What is the power property of logarithms? The power property of logarithms states that if $a$ is a positive real number, then $\log _a x^b = b \log _a x$.
• How do I apply the power property to rewrite a logarithmic expression? To apply the power property, simply multiply the power by the logarithm of the base. For example, if we have $\log _2 x^4$, we can rewrite it as $4 \log _2 x$ using the power property.
• Why is the power property important in mathematics? The power property is important in mathematics because it allows us to rewrite logarithmic expressions in a simpler form. This makes it easier to solve various mathematical problems and understand complex concepts.

Additional Resources

• Logarithmic Expressions: A Comprehensive Guide This article provides a comprehensive guide to logarithmic expressions, including the power property and how to apply it.
• Mathematics: A Step-by-Step Guide This article provides a step-by-step guide to mathematics, including logarithmic expressions and the power property.
• Logarithmic Expressions: Practice Problems This article provides practice problems on logarithmic expressions, including the power property and how to apply it.
  Logarithmic Expressions: A Q&A Guide =====================================

Introduction

Logarithmic expressions are a fundamental concept in mathematics, and understanding how to work with them is crucial for solving various mathematical problems. In this article, we will provide a Q&A guide to logarithmic expressions, including the power property and how to apply it.

Q: What is the power property of logarithms?

A: The power property of logarithms states that if $a$ is a positive real number, then $\log _a x^b = b \log _a x$. This means that when we have a logarithmic expression with a base raised to a power, we can rewrite it as the power multiplied by the logarithm of the base.

Q: How do I apply the power property to rewrite a logarithmic expression?

A: To apply the power property, simply multiply the power by the logarithm of the base. For example, if we have $\log _2 x^4$, we can rewrite it as $4 \log _2 x$ using the power property.

Q: Why is the power property important in mathematics?

A: The power property is important in mathematics because it allows us to rewrite logarithmic expressions in a simpler form. This makes it easier to solve various mathematical problems and understand complex concepts.

Q: What are some common mistakes to avoid when working with logarithmic expressions?

A: Some common mistakes to avoid when working with logarithmic expressions include:

• Not applying the power property correctly: Make sure to multiply the power by the logarithm of the base when applying the power property.
• Not checking the domain of the logarithmic expression: Make sure that the base of the logarithmic expression is positive and the argument is positive.
• Not simplifying the expression: Make sure to simplify the expression as much as possible before solving it.

Q: How do I simplify a logarithmic expression?

A: To simplify a logarithmic expression, follow these steps:

1. Apply the power property: If the expression has a base raised to a power, apply the power property to rewrite it as the power multiplied by the logarithm of the base.
2. Combine like terms: Combine any like terms in the expression.
3. Simplify the expression: Simplify the expression as much as possible.

Q: What are some common logarithmic expressions that can be simplified using the power property?

A: Some common logarithmic expressions that can be simplified using the power property include:

• $\log _2 x^4 = 4 \log _2 x$
• $\log _3 x^5 = 5 \log _3 x$
• $\log _4 x^2 = 2 \log _4 x$

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, follow these steps:

1. Isolate the logarithmic expression: Isolate the logarithmic expression on one side of the equation.
2. Apply the power property: If the expression has a base raised to a power, apply the power property to rewrite it as the power multiplied by the logarithm of the base.
3. Solve for the variable: Solve for the variable by equating the logarithmic expression to the argument.

Q: What are some common logarithmic equations that can be solved using the power property?

A: Some common logarithmic equations that can be solved using the power property include:

• $\log _2 x^4 = 3$
• $\log _3 x^5 = 2$
• $\log _4 x^2 = 1$

Conclusion

In conclusion, logarithmic expressions are a fundamental concept in mathematics, and understanding how to work with them is crucial for solving various mathematical problems. In this article, we have provided a Q&A guide to logarithmic expressions, including the power property and how to apply it. By following this guide, you should be able to work with logarithmic expressions with ease and solve various mathematical problems.

Frequently Asked Questions

• What is the power property of logarithms? The power property of logarithms states that if $a$ is a positive real number, then $\log _a x^b = b \log _a x$.
• How do I apply the power property to rewrite a logarithmic expression? To apply the power property, simply multiply the power by the logarithm of the base. For example, if we have $\log _2 x^4$, we can rewrite it as $4 \log _2 x$ using the power property.
• Why is the power property important in mathematics? The power property is important in mathematics because it allows us to rewrite logarithmic expressions in a simpler form. This makes it easier to solve various mathematical problems and understand complex concepts.

Additional Resources

• Logarithmic Expressions: A Comprehensive Guide This article provides a comprehensive guide to logarithmic expressions, including the power property and how to apply it.
• Mathematics: A Step-by-Step Guide This article provides a step-by-step guide to mathematics, including logarithmic expressions and the power property.
• Logarithmic Expressions: Practice Problems This article provides practice problems on logarithmic expressions, including the power property and how to apply it.