Write As A Single Logarithm: ${ 2\left(\log _2(x-1)^3 + 7 \log _2(x+4)^5\right) }$A. { \log _2\left((x-1) {\frac{3}{2}}(x+4) {\frac{5}{2}}\right)$}$B. { \log _2\left((x-1) 3(x+4) 5\right)^2$} C . \[ C. \[ C . \[ \log

Understanding Logarithmic Properties

When dealing with logarithmic expressions, it's essential to understand the properties that govern their behavior. One of the most critical properties is the power rule, which states that $\log_a(b^c) = c \log_a(b)$. This property allows us to simplify complex logarithmic expressions by applying the exponent to the logarithm.

The Problem at Hand

We are given the expression ${ 2\left(\log _2(x-1)^3 + 7 \log _2(x+4)^5\right) }$. Our goal is to simplify this expression and write it as a single logarithm.

Applying the Power Rule

To simplify the given expression, we can start by applying the power rule to each of the logarithmic terms. This will allow us to rewrite the expression in a more manageable form.

{ 2\left(\log _2(x-1)^3 + 7 \log _2(x+4)^5\right) \}

Using the power rule, we can rewrite the expression as:

{ 2\left(3\log _2(x-1) + 7 \cdot 5 \log _2(x+4)\right) \}

Simplifying further, we get:

{ 6\log _2(x-1) + 35\log _2(x+4) \}

Combining Logarithmic Terms

Now that we have simplified the expression, we can combine the logarithmic terms using the product rule, which states that $\log_a(b) + \log_a(c) = \log_a(bc)$. This will allow us to rewrite the expression as a single logarithm.

{ 6\log _2(x-1) + 35\log _2(x+4) \}

Using the product rule, we can rewrite the expression as:

{ \log _2((x-1)^6(x+4)^{35}) \}

Applying the Power Rule Again

Finally, we can apply the power rule again to simplify the expression further.

{ \log _2((x-1)^6(x+4)^{35}) \}

Using the power rule, we can rewrite the expression as:

{ \log _2((x-1)^6(x+4)^{35}) = \log _2((x-1)^{6 \cdot 2}(x+4)^{35 \cdot 2}) \}

Simplifying further, we get:

{ \log _2((x-1)^{12}(x+4)^{70}) \}

Conclusion

In conclusion, we have successfully simplified the given logarithmic expression and written it as a single logarithm. The final expression is:

{ \log _2((x-1)^{12}(x+4)^{70}) \}

This expression is in the form of a single logarithm, and it meets the requirements of the problem.

Answer Options

Now that we have simplified the expression, let's compare it to the answer options provided.

A. ${ \log _2\left((x-1)^{{\frac{3}{2}}(x+4)}{\frac{5}{2}}\right) }$

B. ${ \log _2\left((x-1)^3(x+4)5\right)^2 }$

C. ${ \log _2\left((x-1)^{12}(x+4){70}\right) }$

Based on our simplification, we can see that option C is the correct answer.

Final Answer

The final answer is:

$\boxed{C}$

Understanding Logarithmic Properties

When dealing with logarithmic expressions, it's essential to understand the properties that govern their behavior. One of the most critical properties is the power rule, which states that $\log_a(b^c) = c \log_a(b)$. This property allows us to simplify complex logarithmic expressions by applying the exponent to the logarithm.

The Problem at Hand

We are given the expression ${ 2\left(\log _2(x-1)^3 + 7 \log _2(x+4)^5\right) }$. Our goal is to simplify this expression and write it as a single logarithm.

Applying the Power Rule

To simplify the given expression, we can start by applying the power rule to each of the logarithmic terms. This will allow us to rewrite the expression in a more manageable form.

{ 2\left(\log _2(x-1)^3 + 7 \log _2(x+4)^5\right) \}

Using the power rule, we can rewrite the expression as:

{ 2\left(3\log _2(x-1) + 7 \cdot 5 \log _2(x+4)\right) \}

Simplifying further, we get:

{ 6\log _2(x-1) + 35\log _2(x+4) \}

Combining Logarithmic Terms

Now that we have simplified the expression, we can combine the logarithmic terms using the product rule, which states that $\log_a(b) + \log_a(c) = \log_a(bc)$. This will allow us to rewrite the expression as a single logarithm.

{ 6\log _2(x-1) + 35\log _2(x+4) \}

Using the product rule, we can rewrite the expression as:

{ \log _2((x-1)^6(x+4)^{35}) \}

Applying the Power Rule Again

Finally, we can apply the power rule again to simplify the expression further.

{ \log _2((x-1)^6(x+4)^{35}) \}

Using the power rule, we can rewrite the expression as:

{ \log _2((x-1)^{6 \cdot 2}(x+4)^{35 \cdot 2}) \}

Simplifying further, we get:

{ \log _2((x-1)^{12}(x+4)^{70}) \}

Conclusion

In conclusion, we have successfully simplified the given logarithmic expression and written it as a single logarithm. The final expression is:

{ \log _2((x-1)^{12}(x+4)^{70}) \}

This expression is in the form of a single logarithm, and it meets the requirements of the problem.

Answer Options

Now that we have simplified the expression, let's compare it to the answer options provided.

A. ${ \log _2\left((x-1)^{{\frac{3}{2}}(x+4)}{\frac{5}{2}}\right) }$

B. ${ \log _2\left((x-1)^3(x+4)5\right)^2 }$

C. ${ \log _2\left((x-1)^{12}(x+4){70}\right) }$

Based on our simplification, we can see that option C is the correct answer.

Final Answer

The final answer is:

$\boxed{C}$

Q&A

Q: What is the power rule for logarithms?

A: The power rule for logarithms states that $\log_a(b^c) = c \log_a(b)$. This property allows us to simplify complex logarithmic expressions by applying the exponent to the logarithm.

Q: How do I simplify a logarithmic expression with multiple terms?

A: To simplify a logarithmic expression with multiple terms, you can use the product rule, which states that $\log_a(b) + \log_a(c) = \log_a(bc)$. This will allow you to rewrite the expression as a single logarithm.

Q: What is the product rule for logarithms?

A: The product rule for logarithms states that $\log_a(b) + \log_a(c) = \log_a(bc)$. This property allows us to combine logarithmic terms and rewrite the expression as a single logarithm.

Q: How do I apply the power rule to a logarithmic expression?

A: To apply the power rule to a logarithmic expression, you can multiply the exponent by the logarithm. For example, if you have the expression $\log_a(b^c)$, you can rewrite it as $c \log_a(b)$.

Q: What is the final answer to the problem?

A: The final answer to the problem is $\boxed{C}$, which is the expression $\log _2\left((x-1)^{12}(x+4)^{70}\right)$.

Q: Why is it important to simplify logarithmic expressions?

A: Simplifying logarithmic expressions is important because it allows us to rewrite complex expressions in a more manageable form. This can make it easier to solve problems and understand the underlying mathematics.

Q: Can you provide more examples of simplifying logarithmic expressions?

A: Yes, here are a few more examples:

• $\log_a(b^c) = c \log_a(b)$
• $\log_a(b) + \log_a(c) = \log_a(bc)$
• $\log_a(b^c) + \log_a(d^e) = \log_a(b^c \cdot d^e)$

These are just a few examples of the many properties and rules that govern logarithmic expressions. By understanding these properties and rules, you can simplify complex logarithmic expressions and solve problems more easily.