Power Property: Log ⁡ B X R = R Log ⁡ B X \log _b X^r = R \log _b X Lo G B ​ X R = R Lo G B ​ X Which Of The Following Is Equivalent To Log ⁡ 3 8 \log _3 8 Lo G 3 ​ 8 ?A. Log ⁡ 6 3 \log _6 3 Lo G 6 ​ 3 B. 2 Log ⁡ 3 3 2 \log _3 3 2 Lo G 3 ​ 3 C. 3 Log ⁡ 3 2 3 \log _3 2 3 Lo G 3 ​ 2 D. 2

Introduction

In the world of mathematics, logarithms play a crucial role in solving various problems, particularly those involving exponential equations. One of the fundamental properties of logarithms is the power property, which states that $\log _b x^r = r \log _b x$. This property allows us to simplify complex logarithmic expressions and solve problems more efficiently. In this article, we will explore the power property of logarithms and apply it to a specific problem to find the equivalent expression.

Understanding the Power Property

The power property of logarithms is a fundamental concept in mathematics that helps us simplify complex logarithmic expressions. It states that $\log _b x^r = r \log _b x$, where $b$ is the base of the logarithm, $x$ is the argument, and $r$ is the exponent. This property can be applied to both base-10 and base-$e$ logarithms.

To understand the power property, let's consider an example. Suppose we want to find the logarithm of $x^r$ with base $b$. We can rewrite $x^r$ as $(x^r)^1$, which is equivalent to $x^{r \cdot 1}$. Using the power property, we can rewrite this expression as $r \log _b x$.

Applying the Power Property to a Problem

Now that we have a good understanding of the power property, let's apply it to a specific problem. We are given the expression $\log _3 8$ and asked to find an equivalent expression. To do this, we can use the power property to rewrite the expression.

First, let's rewrite $8$ as $2^3$. This gives us $\log _3 2^3$. Using the power property, we can rewrite this expression as $3 \log _3 2$.

Therefore, the equivalent expression for $\log _3 8$ is $3 \log _3 2$.

Comparing the Options

Now that we have found the equivalent expression for $\log _3 8$, let's compare it to the options given.

Option A: $\log _6 3$ Option B: $2 \log _3 3$ Option C: $3 \log _3 2$ Option D: 2

Comparing the options, we can see that only one of them is equivalent to $3 \log _3 2$.

Conclusion

In conclusion, the power property of logarithms is a fundamental concept in mathematics that helps us simplify complex logarithmic expressions. By applying this property, we can rewrite expressions in a more manageable form and solve problems more efficiently. In this article, we applied the power property to a specific problem and found the equivalent expression for $\log _3 8$. We also compared the options given and found that only one of them is equivalent to the expression we found.

Key Takeaways

• The power property of logarithms states that $\log _b x^r = r \log _b x$.
• This property can be applied to both base-10 and base-$e$ logarithms.
• The power property can be used to simplify complex logarithmic expressions.
• By applying the power property, we can rewrite expressions in a more manageable form and solve problems more efficiently.

Final Answer

Introduction

In our previous article, we explored the power property of logarithms, which states that $\log _b x^r = r \log _b x$. This property is a fundamental concept in mathematics that helps us simplify complex logarithmic expressions and solve problems more efficiently. In this article, we will answer some frequently asked questions about the power property of logarithms.

Q&A

Q: What is the power property of logarithms?

A: The power property of logarithms states that $\log _b x^r = r \log _b x$, where $b$ is the base of the logarithm, $x$ is the argument, and $r$ is the exponent.

Q: How do I apply the power property of logarithms?

A: To apply the power property of logarithms, you can rewrite the expression $\log _b x^r$ as $r \log _b x$. This can be done by using the property $\log _b x^r = \log _b (x^r)$, which is equivalent to $r \log _b x$.

Q: What are some common applications of the power property of logarithms?

A: The power property of logarithms has many applications in mathematics, including:

• Simplifying complex logarithmic expressions
• Solving exponential equations
• Finding the logarithm of a power
• Converting between different bases of logarithms

Q: Can I use the power property of logarithms with base-10 logarithms?

A: Yes, the power property of logarithms can be applied to both base-10 and base-$e$ logarithms. The property $\log _b x^r = r \log _b x$ holds true for any base $b$.

Q: How do I use the power property of logarithms to simplify an expression?

A: To simplify an expression using the power property of logarithms, you can follow these steps:

1. Identify the base and the argument of the logarithm.
2. Identify the exponent of the argument.
3. Rewrite the expression using the power property of logarithms.
4. Simplify the resulting expression.

Q: Can I use the power property of logarithms to find the logarithm of a power?

A: Yes, the power property of logarithms can be used to find the logarithm of a power. For example, if you want to find the logarithm of $x^r$, you can use the power property to rewrite the expression as $r \log _b x$.

Q: What are some common mistakes to avoid when using the power property of logarithms?

A: Some common mistakes to avoid when using the power property of logarithms include:

• Forgetting to rewrite the expression using the power property of logarithms
• Not identifying the base and the argument of the logarithm
• Not identifying the exponent of the argument
• Not simplifying the resulting expression

Conclusion

In conclusion, the power property of logarithms is a fundamental concept in mathematics that helps us simplify complex logarithmic expressions and solve problems more efficiently. By understanding the power property and how to apply it, you can become more proficient in solving logarithmic problems. We hope this Q&A article has been helpful in answering some of your questions about the power property of logarithms.

Key Takeaways

• The power property of logarithms states that $\log _b x^r = r \log _b x$.
• This property can be applied to both base-10 and base-$e$ logarithms.
• The power property can be used to simplify complex logarithmic expressions.
• By applying the power property, you can rewrite expressions in a more manageable form and solve problems more efficiently.

Final Answer

The final answer is $\boxed{C}$, which is equivalent to $3 \log _3 2$.