Expand Each Expression.A. Ln ⁡ ( 2 X ) 4 \ln (2x)^4 Ln ( 2 X ) 4 B. 4 Ln ⁡ 2 + 4 Ln ⁡ X 4 \ln 2 + 4 \ln X 4 Ln 2 + 4 Ln X C. 4 Ln ⁡ 2 + Ln ⁡ X 4 \ln 2 + \ln X 4 Ln 2 + Ln X D. 8 Ln ⁡ X 8 \ln X 8 Ln X

Introduction

Logarithmic expressions are a fundamental concept in mathematics, and expanding them is a crucial skill to master. In this article, we will explore the process of expanding logarithmic expressions, focusing on the properties of logarithms. We will use the given expressions as examples to demonstrate the step-by-step process of expanding logarithmic expressions.

A. Expanding $\ln (2x)^4$

To expand the expression $\ln (2x)^4$, we need to apply the property of logarithms that states $\log_a (b^c) = c \log_a b$. In this case, we have $\ln (2x)^4$, which can be rewritten as $4 \ln (2x)$.

$\ln (2x)^4 = 4 \ln (2x)$

Next, we can apply the property of logarithms that states $\log_a (bc) = \log_a b + \log_a c$. In this case, we have $4 \ln (2x)$, which can be rewritten as $4 (\ln 2 + \ln x)$.

$4 \ln (2x) = 4 (\ln 2 + \ln x)$

Using the property of logarithms that states $\log_a b + \log_a c = \log_a (bc)$, we can rewrite $4 (\ln 2 + \ln x)$ as $4 \ln 2 + 4 \ln x$.

$4 (\ln 2 + \ln x) = 4 \ln 2 + 4 \ln x$

Therefore, the expanded form of $\ln (2x)^4$ is $4 \ln 2 + 4 \ln x$.

B. Expanding $4 \ln 2 + 4 \ln x$

To expand the expression $4 \ln 2 + 4 \ln x$, we need to apply the property of logarithms that states $\log_a b + \log_a c = \log_a (bc)$. In this case, we have $4 \ln 2 + 4 \ln x$, which can be rewritten as $4 (\ln 2 + \ln x)$.

$4 \ln 2 + 4 \ln x = 4 (\ln 2 + \ln x)$

Using the property of logarithms that states $\log_a b + \log_a c = \log_a (bc)$, we can rewrite $4 (\ln 2 + \ln x)$ as $\ln (2x)^4$.

$4 (\ln 2 + \ln x) = \ln (2x)^4$

Therefore, the expanded form of $4 \ln 2 + 4 \ln x$ is $\ln (2x)^4$.

C. Expanding $4 \ln 2 + \ln x$

To expand the expression $4 \ln 2 + \ln x$, we need to apply the property of logarithms that states $\log_a b + \log_a c = \log_a (bc)$. In this case, we have $4 \ln 2 + \ln x$, which can be rewritten as $\ln 2^4 + \ln x$.

$4 \ln 2 + \ln x = \ln 2^4 + \ln x$

Using the property of logarithms that states $\log_a b + \log_a c = \log_a (bc)$, we can rewrite $\ln 2^4 + \ln x$ as $\ln (2^4x)$.

$\ln 2^4 + \ln x = \ln (2^4x)$

Using the property of logarithms that states $\log_a b^c = c \log_a b$, we can rewrite $\ln (2^4x)$ as $4 \ln 2 + \ln x$.

$\ln (2^4x) = 4 \ln 2 + \ln x$

Therefore, the expanded form of $4 \ln 2 + \ln x$ is $4 \ln 2 + \ln x$.

D. Expanding $8 \ln x$

To expand the expression $8 \ln x$, we need to apply the property of logarithms that states $\log_a b^c = c \log_a b$. In this case, we have $8 \ln x$, which can be rewritten as $\ln x^8$.

$8 \ln x = \ln x^8$

Using the property of logarithms that states $\log_a b^c = c \log_a b$, we can rewrite $\ln x^8$ as $8 \ln x$.

$\ln x^8 = 8 \ln x$

Therefore, the expanded form of $8 \ln x$ is $8 \ln x$.

Conclusion

Introduction

Logarithmic expressions are a fundamental concept in mathematics, and understanding them is crucial for success in various fields. In this article, we will address some of the most frequently asked questions about logarithmic expressions, providing clear and concise answers to help you better understand this complex topic.

Q: What is a logarithmic expression?

A: A logarithmic expression is an expression that involves a logarithm, which is the inverse operation of exponentiation. In other words, a logarithmic expression is a way of expressing a number in terms of its logarithm.

Q: What are the properties of logarithms?

A: The properties of logarithms are a set of rules that govern how logarithms behave. The main properties of logarithms are:

• Product Rule: $\log_a (bc) = \log_a b + \log_a c$
• Quotient Rule: $\log_a \frac{b}{c} = \log_a b - \log_a c$
• Power Rule: $\log_a b^c = c \log_a b$
• Change of Base Rule: $\log_b a = \frac{\log_c a}{\log_c b}$

Q: How do I expand a logarithmic expression?

A: To expand a logarithmic expression, you need to apply the properties of logarithms. Here are the steps to follow:

1. Identify the logarithmic expression you want to expand.
2. Apply the product rule to expand the expression.
3. Apply the power rule to simplify the expression.
4. Apply the change of base rule to change the base of the logarithm.

Q: What is the difference between a logarithmic expression and an exponential expression?

A: A logarithmic expression is an expression that involves a logarithm, while an exponential expression is an expression that involves an exponent. In other words, a logarithmic expression is a way of expressing a number in terms of its logarithm, while an exponential expression is a way of expressing a number in terms of its exponent.

Q: How do I simplify a logarithmic expression?

A: To simplify a logarithmic expression, you need to apply the properties of logarithms. Here are the steps to follow:

1. Identify the logarithmic expression you want to simplify.
2. Apply the product rule to simplify the expression.
3. Apply the power rule to simplify the expression.
4. Apply the change of base rule to change the base of the logarithm.

Q: What is the logarithmic form of a number?

A: The logarithmic form of a number is a way of expressing the number in terms of its logarithm. For example, the logarithmic form of 100 is $\log_{10} 100$, which is equal to 2.

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

A: To convert a logarithmic expression to exponential form, you need to apply the inverse operation of logarithms, which is exponentiation. Here are the steps to follow:

1. Identify the logarithmic expression you want to convert.
2. Apply the change of base rule to change the base of the logarithm.
3. Apply the power rule to simplify the expression.
4. Apply the exponential function to convert the expression to exponential form.

Conclusion

In this article, we have addressed some of the most frequently asked questions about logarithmic expressions, providing clear and concise answers to help you better understand this complex topic. By applying the properties of logarithms and following the steps outlined in this article, you can expand, simplify, and convert logarithmic expressions with ease.