Given $\log _4 3 \approx 0.792$ And $\log _4 21 \approx 2.196$, What Is $\log _4 7$?A. 1.404 B. 1.739 C. 2.773 D. 2.988

Introduction

Logarithms are a fundamental concept in mathematics, used to represent the power to which a base number must be raised to obtain a given value. In this article, we will explore the properties of logarithms and use them to find the value of $\log _4 7$ given the approximations of $\log _4 3 \approx 0.792$ and $\log _4 21 \approx 2.196$.

Logarithmic Properties

Before we dive into the problem, let's review some essential logarithmic properties:

• Product Rule: $\log _b (xy) = \log _b x + \log _b y$
• Quotient Rule: $\log _b \left(\frac{x}{y}\right) = \log _b x - \log _b y$
• Power Rule: $\log _b x^y = y \log _b x$

These properties will be crucial in solving the problem.

The Problem

We are given the approximations of $\log _4 3 \approx 0.792$ and $\log _4 21 \approx 2.196$. Our goal is to find the value of $\log _4 7$.

Step 1: Expressing 21 as a Product of 3 and 7

We can express 21 as a product of 3 and 7: $21 = 3 \cdot 7$. Using the product rule, we can write:

$$\log _4 21 = \log _4 (3 \cdot 7) = \log _4 3 + \log _4 7$$

Step 2: Substituting the Given Approximations

We can substitute the given approximations into the equation:

$$2.196 = 0.792 + \log _4 7$$

Step 3: Solving for $\log _4 7$

To solve for $\log _4 7$, we need to isolate it on one side of the equation. We can do this by subtracting 0.792 from both sides:

$$\log _4 7 = 2.196 - 0.792$$

Step 4: Evaluating the Expression

Now, we can evaluate the expression:

$$\log _4 7 = 1.404$$

Conclusion

Using the product rule and the given approximations, we were able to find the value of $\log _4 7$. The final answer is $\boxed{1.404}$.

Discussion

This problem demonstrates the power of logarithmic properties in solving problems. By using the product rule, we were able to express 21 as a product of 3 and 7, and then use the given approximations to find the value of $\log _4 7$. This type of problem is essential in mathematics, as it helps us develop problem-solving skills and understand the properties of logarithms.

Real-World Applications

Logarithms have numerous real-world applications, including:

• Finance: Logarithms are used to calculate interest rates, investment returns, and risk analysis.
• Science: Logarithms are used to measure the intensity of earthquakes, the brightness of stars, and the concentration of chemicals.
• Engineering: Logarithms are used to design electronic circuits, calculate signal processing, and optimize system performance.

In conclusion, logarithms are a fundamental concept in mathematics, and their properties are essential in solving problems. By understanding logarithmic properties and using them to find the value of $\log _4 7$, we can develop problem-solving skills and apply logarithms to real-world applications.

Additional Resources

For further learning, we recommend the following resources:

• Mathematics textbooks: "Calculus" by Michael Spivak, "Algebra" by Michael Artin
• Online resources: Khan Academy, MIT OpenCourseWare, Wolfram Alpha
• Mathematical software: Mathematica, Maple, MATLAB

Introduction

In our previous article, we explored the properties of logarithms and used them to find the value of $\log _4 7$ given the approximations of $\log _4 3 \approx 0.792$ and $\log _4 21 \approx 2.196$. In this article, we will answer some frequently asked questions related to logarithmic properties and approximations.

Q&A

Q: What is the product rule of logarithms?

A: The product rule of logarithms states that $\log _b (xy) = \log _b x + \log _b y$. This means that the logarithm of a product is equal to the sum of the logarithms of the individual factors.

Q: How do I use the product rule to find the value of $\log _4 21$?

A: To find the value of $\log _4 21$, we can express 21 as a product of 3 and 7: $21 = 3 \cdot 7$. Using the product rule, we can write:

$$\log _4 21 = \log _4 (3 \cdot 7) = \log _4 3 + \log _4 7$$

Q: What is the quotient rule of logarithms?

A: The quotient rule of logarithms states that $\log _b \left(\frac{x}{y}\right) = \log _b x - \log _b y$. This means that the logarithm of a quotient is equal to the difference of the logarithms of the individual factors.

Q: How do I use the quotient rule to find the value of $\log _4 \left(\frac{21}{3}\right)$?

A: To find the value of $\log _4 \left(\frac{21}{3}\right)$, we can use the quotient rule:

$$\log _4 \left(\frac{21}{3}\right) = \log _4 21 - \log _4 3$$

Q: What is the power rule of logarithms?

A: The power rule of logarithms states that $\log _b x^y = y \log _b x$. This means that the logarithm of a power is equal to the exponent multiplied by the logarithm of the base.

Q: How do I use the power rule to find the value of $\log _4 7^2$?

A: To find the value of $\log _4 7^2$, we can use the power rule:

$$\log _4 7^2 = 2 \log _4 7$$

Q: How do I use logarithmic properties to find the value of $\log _4 7$?

A: To find the value of $\log _4 7$, we can use the product rule and the given approximations:

$$\log _4 21 = \log _4 (3 \cdot 7) = \log _4 3 + \log _4 7$$

$$2.196 = 0.792 + \log _4 7$$

$$\log _4 7 = 2.196 - 0.792$$

$$\log _4 7 = 1.404$$

Conclusion

In this article, we answered some frequently asked questions related to logarithmic properties and approximations. By understanding these properties and using them to find the value of $\log _4 7$, we can develop problem-solving skills and apply logarithms to real-world applications.

Additional Resources

For further learning, we recommend the following resources:

• Mathematics textbooks: "Calculus" by Michael Spivak, "Algebra" by Michael Artin
• Online resources: Khan Academy, MIT OpenCourseWare, Wolfram Alpha
• Mathematical software: Mathematica, Maple, MATLAB

By exploring these resources, you can deepen your understanding of logarithmic properties and apply them to real-world problems.

Frequently Asked Questions

Q: What is the difference between a logarithm and an exponent?

A: A logarithm is the inverse operation of an exponent. While an exponent represents the power to which a base number must be raised to obtain a given value, a logarithm represents the power to which a base number must be raised to obtain a given value.

Q: How do I convert a logarithmic expression to an exponential expression?

A: To convert a logarithmic expression to an exponential expression, you can use the definition of a logarithm:

$$\log _b x = y \iff b^y = x$$

Q: What is the relationship between logarithms and exponents?

A: Logarithms and exponents are inverse operations. While an exponent represents the power to which a base number must be raised to obtain a given value, a logarithm represents the power to which a base number must be raised to obtain a given value.

Conclusion

In conclusion, logarithmic properties and approximations are essential in mathematics and have numerous real-world applications. By understanding these properties and using them to find the value of $\log _4 7$, we can develop problem-solving skills and apply logarithms to real-world problems.