Product Property: ${ G_b(xy) = \log_b X + \log_b Y }$How Would You Expand { \log_4 12$}$ So That It Can Be Evaluated, Given { \log_4 3 \approx 0.792$}$?A. { \log_4 3 \cdot \log_4 4 \sqrt{a 2+b 2}$}$ B.

Introduction

Logarithmic functions are a fundamental concept in mathematics, and understanding their properties is crucial for solving various mathematical problems. In this article, we will explore the property of logarithms that allows us to expand and evaluate logarithmic expressions. Specifically, we will focus on the property $g_b(xy) = \log_b x + \log_b y$ and use it to expand the expression $\log_4 12$ given that $\log_4 3 \approx 0.792$.

Understanding Logarithmic Properties

Before we dive into the expansion of the expression $\log_4 12$, let's briefly review the logarithmic properties that we will be using. The property $g_b(xy) = \log_b x + \log_b y$ states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. This property can be rewritten as $\log_b (xy) = \log_b x + \log_b y$.

Expanding the Expression $\log_4 12$

To expand the expression $\log_4 12$, we can use the property $g_b(xy) = \log_b x + \log_b y$. Since $12 = 3 \cdot 4$, we can rewrite the expression as $\log_4 (3 \cdot 4)$. Using the property, we can expand this expression as $\log_4 3 + \log_4 4$.

Evaluating the Expression $\log_4 3 + \log_4 4$

Now that we have expanded the expression $\log_4 12$ as $\log_4 3 + \log_4 4$, we can evaluate it using the given value of $\log_4 3 \approx 0.792$. Since $\log_4 4 = 1$, we can substitute this value into the expression to get $\log_4 3 + 1$.

Simplifying the Expression $\log_4 3 + 1$

To simplify the expression $\log_4 3 + 1$, we can use the fact that $\log_b b = 1$. Since $\log_4 4 = 1$, we can rewrite the expression as $\log_4 3 + \log_4 4^1$. Using the property $g_b(xy) = \log_b x + \log_b y$, we can expand this expression as $\log_4 (3 \cdot 4^1)$.

Evaluating the Expression $\log_4 (3 \cdot 4^1)$

Now that we have simplified the expression $\log_4 3 + 1$ as $\log_4 (3 \cdot 4^1)$, we can evaluate it using the given value of $\log_4 3 \approx 0.792$. Since $4^1 = 4$, we can substitute this value into the expression to get $\log_4 (3 \cdot 4)$.

Using the Property $g_b(xy) = \log_b x + \log_b y$

To evaluate the expression $\log_4 (3 \cdot 4)$, we can use the property $g_b(xy) = \log_b x + \log_b y$. Since $3 \cdot 4 = 12$, we can rewrite the expression as $\log_4 12$. Using the given value of $\log_4 3 \approx 0.792$, we can substitute this value into the expression to get $\log_4 12 \approx 0.792 + \log_4 4$.

Evaluating the Expression $\log_4 12 \approx 0.792 + \log_4 4$

Now that we have evaluated the expression $\log_4 12 \approx 0.792 + \log_4 4$, we can simplify it using the fact that $\log_b b = 1$. Since $\log_4 4 = 1$, we can rewrite the expression as $\log_4 12 \approx 0.792 + 1$.

Conclusion

In this article, we have explored the property of logarithms that allows us to expand and evaluate logarithmic expressions. Specifically, we have used the property $g_b(xy) = \log_b x + \log_b y$ to expand the expression $\log_4 12$ given that $\log_4 3 \approx 0.792$. We have shown that the expanded expression can be evaluated using the given value of $\log_4 3 \approx 0.792$ and the fact that $\log_b b = 1$. The final answer is $\boxed{1.792}$.

Answer Key

A. $\log_4 3 \cdot \log_4 4 \sqrt{a^2+b^2}$ is not the correct answer.

The correct answer is $\boxed{1.792}$.

Discussion

The property $g_b(xy) = \log_b x + \log_b y$ is a fundamental concept in mathematics that allows us to expand and evaluate logarithmic expressions. In this article, we have shown how to use this property to expand the expression $\log_4 12$ given that $\log_4 3 \approx 0.792$. We have also shown how to evaluate the expanded expression using the given value of $\log_4 3 \approx 0.792$ and the fact that $\log_b b = 1$.

References

• [1] "Logarithmic Properties" by Math Open Reference
• [2] "Logarithmic Functions" by Khan Academy

Related Articles

• "Logarithmic Properties: Simplifying Logarithmic Expressions"
• "Logarithmic Functions: Evaluating Logarithmic Expressions"
• "Mathematical Properties: Understanding Logarithmic Functions"
  Logarithmic Properties: Q&A =============================

Q: What is the property of logarithms that allows us to expand and evaluate logarithmic expressions?

A: The property of logarithms that allows us to expand and evaluate logarithmic expressions is $g_b(xy) = \log_b x + \log_b y$. This property states that the logarithm of a product is equal to the sum of the logarithms of the individual factors.

Q: How do we use the property $g_b(xy) = \log_b x + \log_b y$ to expand the expression $\log_4 12$?

A: To expand the expression $\log_4 12$, we can use the property $g_b(xy) = \log_b x + \log_b y$. Since $12 = 3 \cdot 4$, we can rewrite the expression as $\log_4 (3 \cdot 4)$. Using the property, we can expand this expression as $\log_4 3 + \log_4 4$.

Q: How do we evaluate the expression $\log_4 3 + \log_4 4$?

A: To evaluate the expression $\log_4 3 + \log_4 4$, we can use the given value of $\log_4 3 \approx 0.792$. Since $\log_4 4 = 1$, we can substitute this value into the expression to get $\log_4 3 + 1$.

Q: How do we simplify the expression $\log_4 3 + 1$?

A: To simplify the expression $\log_4 3 + 1$, we can use the fact that $\log_b b = 1$. Since $\log_4 4 = 1$, we can rewrite the expression as $\log_4 3 + \log_4 4^1$. Using the property $g_b(xy) = \log_b x + \log_b y$, we can expand this expression as $\log_4 (3 \cdot 4^1)$.

Q: How do we evaluate the expression $\log_4 (3 \cdot 4^1)$?

A: To evaluate the expression $\log_4 (3 \cdot 4^1)$, we can use the given value of $\log_4 3 \approx 0.792$. Since $4^1 = 4$, we can substitute this value into the expression to get $\log_4 (3 \cdot 4)$.

Q: How do we use the property $g_b(xy) = \log_b x + \log_b y$ to evaluate the expression $\log_4 (3 \cdot 4)$?

A: To evaluate the expression $\log_4 (3 \cdot 4)$, we can use the property $g_b(xy) = \log_b x + \log_b y$. Since $3 \cdot 4 = 12$, we can rewrite the expression as $\log_4 12$. Using the given value of $\log_4 3 \approx 0.792$, we can substitute this value into the expression to get $\log_4 12 \approx 0.792 + \log_4 4$.

Q: How do we evaluate the expression $\log_4 12 \approx 0.792 + \log_4 4$?

A: To evaluate the expression $\log_4 12 \approx 0.792 + \log_4 4$, we can use the fact that $\log_b b = 1$. Since $\log_4 4 = 1$, we can substitute this value into the expression to get $\log_4 12 \approx 0.792 + 1$.

Q: What is the final answer to the expression $\log_4 12 \approx 0.792 + 1$?

A: The final answer to the expression $\log_4 12 \approx 0.792 + 1$ is $\boxed{1.792}$.

Q: What is the property of logarithms that allows us to simplify logarithmic expressions?

A: The property of logarithms that allows us to simplify logarithmic expressions is $g_b(xy) = \log_b x + \log_b y$. This property states that the logarithm of a product is equal to the sum of the logarithms of the individual factors.

Q: How do we use the property $g_b(xy) = \log_b x + \log_b y$ to simplify the expression $\log_4 12$?

A: To simplify the expression $\log_4 12$, we can use the property $g_b(xy) = \log_b x + \log_b y$. Since $12 = 3 \cdot 4$, we can rewrite the expression as $\log_4 (3 \cdot 4)$. Using the property, we can simplify this expression as $\log_4 3 + \log_4 4$.

Q: How do we evaluate the expression $\log_4 3 + \log_4 4$?

A: To evaluate the expression $\log_4 3 + \log_4 4$, we can use the given value of $\log_4 3 \approx 0.792$. Since $\log_4 4 = 1$, we can substitute this value into the expression to get $\log_4 3 + 1$.

Q: What is the final answer to the expression $\log_4 3 + 1$?

A: The final answer to the expression $\log_4 3 + 1$ is $\boxed{1.792}$.

Conclusion

In this Q&A article, we have explored the property of logarithms that allows us to expand and evaluate logarithmic expressions. We have also shown how to use this property to simplify logarithmic expressions. The final answer to the expression $\log_4 12 \approx 0.792 + 1$ is $\boxed{1.792}$.