Substituting the Simplified Logarithm

Oct 13, 2025 by recoveryshake.com 34 views

Introduction

Logarithms are a fundamental concept in mathematics, and they play a crucial role in various mathematical operations. In this article, we will focus on expanding logarithms, specifically the logarithm $\log_4\left(\frac{w}{64}\right)$ . We will explore the different options provided and determine the correct expansion of the given logarithm.

Understanding Logarithms

Before we dive into expanding the logarithm, let's briefly review the concept of logarithms. A logarithm is the inverse operation of exponentiation. In other words, if $a^b = c$ , then $\log_a c = b$ . The logarithm of a number is the exponent to which a base number must be raised to produce that number.

Expanding the Logarithm

Now, let's focus on expanding the logarithm $\log_4\left(\frac{w}{64}\right)$ . To do this, we can use the properties of logarithms, specifically the quotient property, which states that $\log_a \frac{b}{c} = \log_a b - \log_a c$ .

Using this property, we can rewrite the given logarithm as:

\log_4\left(\frac{w}{64}\right) = \log_4 w - \log_4 64

Simplifying the Logarithm

Now, let's simplify the logarithm $\log_4 64$ . To do this, we can use the fact that $64 = 4^3$ . Therefore, we can rewrite the logarithm as:

\log_4 64 = \log_4 4^3 = 3

Substituting the Simplified Logarithm

Now, let's substitute the simplified logarithm back into the original expression:

\log_4\left(\frac{w}{64}\right) = \log_4 w - 3

Comparing with the Options

Now, let's compare our result with the options provided:

A. $\log_4(w-3)$ B. $\log_4\left(\frac{w}{3}\right)$ C. $\log_4 w - 3$ D. $\frac{\log_4 w}{3}$

Based on our calculation, we can see that the correct expansion of the logarithm is:

C. $\log_4 w - 3$

Conclusion

In this article, we expanded the logarithm $\log_4\left(\frac{w}{64}\right)$ using the properties of logarithms. We simplified the logarithm and substituted the result back into the original expression. Our final answer is:

C. $\log_4 w - 3$

This result is consistent with the properties of logarithms and demonstrates the importance of understanding and applying these properties in mathematical operations.

Additional Tips and Tricks

When working with logarithms, it's essential to remember the following tips and tricks:

Use the properties of logarithms to simplify complex expressions.
Apply the quotient property to rewrite logarithms of fractions.
Use the fact that $a^b = c$ implies $\log_a c = b$ to rewrite logarithms in terms of exponents.
Be careful when simplifying logarithms, as small errors can lead to incorrect results.

By following these tips and tricks, you can become more confident and proficient in working with logarithms and expanding complex expressions.

Common Mistakes to Avoid

When expanding logarithms, it's essential to avoid the following common mistakes:

Failing to apply the properties of logarithms.
Simplifying logarithms incorrectly.
Forgetting to substitute the simplified logarithm back into the original expression.
Not checking the result against the options provided.

By being aware of these common mistakes, you can avoid them and ensure that your results are accurate and reliable.

Real-World Applications

Logarithms have numerous real-world applications, including:

Calculating interest rates and investments.
Analyzing data and trends.
Solving problems in physics, engineering, and computer science.
Modeling population growth and decay.

By understanding and applying logarithms, you can become more proficient in solving complex problems and making informed decisions in various fields.

Conclusion

C. $\log_4 w - 3$

Introduction

In our previous article, we expanded the logarithm $\log_4\left(\frac{w}{64}\right)$ using the properties of logarithms. We simplified the logarithm and substituted the result back into the original expression. In this article, we will provide a Q&A guide to help you better understand and apply logarithms in various mathematical operations.

Q: What is the quotient property of logarithms?

A: The quotient property of logarithms states that $\log_a \frac{b}{c} = \log_a b - \log_a c$ . This property allows us to rewrite logarithms of fractions as the difference of logarithms.

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

A: To apply the quotient property of logarithms, simply rewrite the logarithm of a fraction as the difference of logarithms. For example, $\log_4 \frac{w}{64} = \log_4 w - \log_4 64$ .

Q: What is the fact that $a^b = c$ implies $\log_a c = b$ ?

A: This fact is a fundamental property of logarithms. It states that if $a^b = c$ , then $\log_a c = b$ . This property allows us to rewrite logarithms in terms of exponents.

Q: How do I use the fact that $a^b = c$ implies $\log_a c = b$ ?

A: To use this fact, simply rewrite the logarithm in terms of an exponent. For example, $\log_4 64 = \log_4 4^3 = 3$ .

Q: What is the difference between $\log_4 w - 3$ and $\log_4 w - \log_4 64$ ?

A: The expression $\log_4 w - 3$ is equivalent to $\log_4 w - \log_4 64$ . This is because $3 = \log_4 64$ , so we can substitute this value into the expression.

Q: How do I simplify logarithms?

A: To simplify logarithms, use the properties of logarithms to rewrite the expression in a simpler form. For example, $\log_4 \frac{w}{64} = \log_4 w - \log_4 64 = \log_4 w - 3$ .

Q: What are some common mistakes to avoid when expanding logarithms?

A: Some common mistakes to avoid when expanding logarithms include:

Failing to apply the properties of logarithms.
Simplifying logarithms incorrectly.
Forgetting to substitute the simplified logarithm back into the original expression.
Not checking the result against the options provided.

Q: How do I apply logarithms in real-world applications?

A: Logarithms have numerous real-world applications, including:

Calculating interest rates and investments.
Analyzing data and trends.
Solving problems in physics, engineering, and computer science.
Modeling population growth and decay.

By understanding and applying logarithms, you can become more proficient in solving complex problems and making informed decisions in various fields.

Q: What are some additional tips and tricks for working with logarithms?

A: Some additional tips and tricks for working with logarithms include:

Using the properties of logarithms to simplify complex expressions.
Applying the quotient property to rewrite logarithms of fractions.
Using the fact that $a^b = c$ implies $\log_a c = b$ to rewrite logarithms in terms of exponents.
Being careful when simplifying logarithms, as small errors can lead to incorrect results.

By following these tips and tricks, you can become more confident and proficient in working with logarithms and expanding complex expressions.

Conclusion

In this article, we provided a Q&A guide to help you better understand and apply logarithms in various mathematical operations. We covered topics such as the quotient property of logarithms, simplifying logarithms, and applying logarithms in real-world applications. By following the tips and tricks provided, you can become more confident and proficient in working with logarithms and expanding complex expressions.