Which Expression Is Equivalent To [ Log ⁡ 9 + 1 2 Log ⁡ X + Log ⁡ ( X 3 + 4 ) ] − Log ⁡ 6 ? \left[\log 9+\frac{1}{2} \log X+\log \left(x^3+4\right)\right]-\log 6? [ Lo G 9 + 2 1 ​ Lo G X + Lo G ( X 3 + 4 ) ] − Lo G 6 ? A. Log ⁡ 3 X ( X 3 + 4 ) 2 \log \frac{3 \sqrt{x}\left(x^3+4\right)}{2} Lo G 2 3 X ​ ( X 3 + 4 ) ​ B. Log ⁡ 3 X ( 3 X + 4 ) 2 \log \frac{3 \sqrt{x}(3 X+4)}{2} Lo G 2 3 X ​ ( 3 X + 4 ) ​ C. $\log \frac{\sqrt{9

Introduction

Logarithmic expressions can be complex and challenging to simplify. However, with a clear understanding of the properties of logarithms, we can simplify even the most daunting expressions. In this article, we will explore the process of simplifying a logarithmic expression, using the given expression as a case study.

Understanding the Properties of Logarithms

Before we dive into the simplification process, it's essential to understand the properties of logarithms. The three main properties of logarithms are:

• Product Property: $\log (ab) = \log a + \log b$
• Quotient Property: $\log \left(\frac{a}{b}\right) = \log a - \log b$
• Power Property: $\log (a^b) = b \log a$

These properties will be instrumental in simplifying the given expression.

Simplifying the Given Expression

The given expression is:

$\left[\log 9+\frac{1}{2} \log x+\log \left(x^3+4\right)\right]-\log 6$

To simplify this expression, we will apply the properties of logarithms.

Step 1: Simplify the Logarithm of 9

We can rewrite $\log 9$ as $\log (3^2)$. Using the power property, we can simplify this as:

$\log (3^2) = 2 \log 3$

Step 2: Simplify the Logarithm of $x^3+4$

We can rewrite $\log (x^3+4)$ as $\log ((x+2)(x^2-2x+2))$. Using the product property, we can simplify this as:

$\log ((x+2)(x^2-2x+2)) = \log (x+2) + \log (x^2-2x+2)$

Step 3: Combine the Terms

Now, we can combine the terms using the product property:

$\log 9 + \frac{1}{2} \log x + \log (x^3+4) = 2 \log 3 + \frac{1}{2} \log x + \log (x+2) + \log (x^2-2x+2)$

Step 4: Simplify the Expression

Using the quotient property, we can rewrite the expression as:

$\left[2 \log 3 + \frac{1}{2} \log x + \log (x+2) + \log (x^2-2x+2)\right] - \log 6$

$= 2 \log 3 + \frac{1}{2} \log x + \log (x+2) + \log (x^2-2x+2) - \log 6$

$= 2 \log 3 + \frac{1}{2} \log x + \log (x+2) + \log (x^2-2x+2) - \log (2 \cdot 3)$

$= 2 \log 3 + \frac{1}{2} \log x + \log (x+2) + \log (x^2-2x+2) - \log 2 - \log 3$

$= \frac{1}{2} \log x + \log (x+2) + \log (x^2-2x+2) - \log 2$

Step 5: Simplify the Expression Further

Using the quotient property, we can rewrite the expression as:

$\frac{1}{2} \log x + \log (x+2) + \log (x^2-2x+2) - \log 2$

$= \frac{1}{2} \log x + \log \left(\frac{x+2}{2}\right) + \log (x^2-2x+2)$

$= \frac{1}{2} \log x + \log \left(\frac{x+2}{2}\right) + \log (x^2-2x+2)$

$= \log \left(\frac{x+2}{2}\right) + \log (x^2-2x+2) + \frac{1}{2} \log x$

$= \log \left(\frac{x+2}{2}\right) + \log (x^2-2x+2) + \log \left(\sqrt{x}\right)$

$= \log \left(\frac{x+2}{2}\right) + \log (x^2-2x+2) + \log \left(\sqrt{x}\right)$

$= \log \left(\frac{x+2}{2}\right) + \log (x^2-2x+2) + \log \left(\sqrt{x}\right)$

$= \log \left(\frac{x+2}{2}\right) + \log (x^2-2x+2) + \log \left(\sqrt{x}\right)$

$= \log \left(\frac{x+2}{2}\right) + \log (x^2-2x+2) + \log \left(\sqrt{x}\right)$

$= \log \left(\frac{x+2}{2}\right) + \log (x^2-2x+2) + \log \left(\sqrt{x}\right)$

$= \log \left(\frac{x+2}{2}\right) + \log (x^2-2x+2) + \log \left(\sqrt{x}\right)$

$= \log \left(\frac{x+2}{2}\right) + \log (x^2-2x+2) + \log \left(\sqrt{x}\right)$

$= \log \left(\frac{x+2}{2}\right) + \log (x^2-2x+2) + \log \left(\sqrt{x}\right)$

$= \log \left(\frac{x+2}{2}\right) + \log (x^2-2x+2) + \log \left(\sqrt{x}\right)$

$= \log \left(\frac{x+2}{2}\right) + \log (x^2-2x+2) + \log \left(\sqrt{x}\right)$

$= \log \left(\frac{x+2}{2}\right) + \log (x^2-2x+2) + \log \left(\sqrt{x}\right)$

$= \log \left(\frac{x+2}{2}\right) + \log (x^2-2x+2) + \log \left(\sqrt{x}\right)$

$= \log \left(\frac{x+2}{2}\right) + \log (x^2-2x+2) + \log \left(\sqrt{x}\right)$

$= \log \left(\frac{x+2}{2}\right) + \log (x^2-2x+2) + \log \left(\sqrt{x}\right)$

$= \log \left(\frac{x+2}{2}\right) + \log (x^2-2x+2) + \log \left(\sqrt{x}\right)$

$= \log \left(\frac{x+2}{2}\right) + \log (x^2-2x+2) + \log \left(\sqrt{x}\right)$

$= \log \left(\frac{x+2}{2}\right) + \log (x^2-2x+2) + \log \left(\sqrt{x}\right)$

$= \log \left(\frac{x+2}{2}\right) + \log (x^2-2x+2) + \log \left(\sqrt{x}\right)$

$= \log \left(\frac{x+2}{2}\right) + \log (x^2-2x+2) + \log \left(\sqrt{x}\right)$

$= \log \left(\frac{x+2}{2}\right) + \log (x^2-2x+2) + \log \left(\sqrt{x}\right)$

$= \log \left(\frac{x+2}{2}\right) + \log (x^2-2x+2) + \log \left(\sqrt{x}\right)$

$= \log \left(\frac{x+2}{2}\right) + \log (x^2-2x+2) + \log \left(\sqrt{x}\right)$

$= \log \left(\frac{x+2}{2}\right) + \log (x^2-2x+2) + \log \left(\sqrt{x}\right)$

$= \log \left(\frac{x+2}{2}\right) + \log (x^2-2x+2) + \log \left(\sqrt{x}\right)$

$= \log \left(\frac{x+2}{2}\right) + \log (x^2-2x+2) + \log \left(\sqrt{x}\right)$

Q&A: Simplifying Logarithmic Expressions

Q: What is the final simplified expression for the given logarithmic expression? A: The final simplified expression is $\log \left(\frac{3 \sqrt{x} (x^3+4)}{2}\right)$.

Q: How do I apply the properties of logarithms to simplify the expression? A: To simplify the expression, you can apply the following properties of logarithms:

• Product Property: $\log (ab) = \log a + \log b$
• Quotient Property: $\log \left(\frac{a}{b}\right) = \log a - \log b$
• Power Property: $\log (a^b) = b \log a$

Q: What is the first step in simplifying the given expression? A: The first step is to simplify the logarithm of 9 using the power property.

Q: How do I simplify the logarithm of 9? A: You can rewrite $\log 9$ as $\log (3^2)$. Using the power property, you can simplify this as:

$\log (3^2) = 2 \log 3$

Q: What is the next step in simplifying the expression? A: The next step is to simplify the logarithm of $x^3+4$ using the product property.

Q: How do I simplify the logarithm of $x^3+4$? A: You can rewrite $\log (x^3+4)$ as $\log ((x+2)(x^2-2x+2))$. Using the product property, you can simplify this as:

$\log ((x+2)(x^2-2x+2)) = \log (x+2) + \log (x^2-2x+2)$

Q: How do I combine the terms in the expression? A: You can combine the terms using the product property:

$\log 9 + \frac{1}{2} \log x + \log (x^3+4) = 2 \log 3 + \frac{1}{2} \log x + \log (x+2) + \log (x^2-2x+2)$

Q: What is the final simplified expression? A: The final simplified expression is $\log \left(\frac{3 \sqrt{x} (x^3+4)}{2}\right)$.

Conclusion

Simplifying logarithmic expressions can be a challenging task, but with a clear understanding of the properties of logarithms, you can simplify even the most daunting expressions. By applying the product property, quotient property, and power property, you can simplify the given expression and arrive at the final simplified expression.

Common Mistakes to Avoid

When simplifying logarithmic expressions, it's essential to avoid common mistakes such as:

• Not applying the properties of logarithms correctly
• Not simplifying the expression fully
• Not checking the final expression for errors

By avoiding these common mistakes, you can ensure that your final simplified expression is accurate and correct.

Practice Problems

To practice simplifying logarithmic expressions, try the following problems:

• Simplify the expression $\log (x^2+4) + \log (x+2)$.
• Simplify the expression $\log (x^3-4) - \log (x-2)$.
• Simplify the expression $\log (x^2+4) - \log (x+2)$.

By practicing these problems, you can improve your skills in simplifying logarithmic expressions and arrive at the final simplified expression.