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

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Introduction

Logarithmic expressions can be complex and challenging to simplify. However, with a clear understanding of the properties of logarithms, we can break down these expressions into more manageable parts. In this article, we will explore how to simplify a given logarithmic expression and identify the equivalent expression among the provided options.

Understanding Logarithmic Properties

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

  • Product Property: log⁡(xy)=log⁡x+log⁡y\log(xy) = \log x + \log y
  • Quotient Property: log⁡(xy)=log⁡x−log⁡y\log\left(\frac{x}{y}\right) = \log x - \log y
  • Power Property: log⁡(xy)=ylog⁡x\log(x^y) = y\log x
  • Change of Base Property: log⁡ba=log⁡calog⁡cb\log_b a = \frac{\log_c a}{\log_c b}

These properties will be crucial in simplifying the given expression.

Simplifying the Given Expression

The given expression is:

[log⁡9+12log⁡x+log⁡(x3+4)]−log⁡6\left[\log 9 + \frac{1}{2} \log x + \log \left(x^3+4\right)\right] - \log 6

To simplify this expression, we can start by applying the product property to combine the logarithms:

log⁡9+12log⁡x+log⁡(x3+4)=log⁡(9x12(x3+4))\log 9 + \frac{1}{2} \log x + \log \left(x^3+4\right) = \log(9x^{\frac{1}{2}}(x^3+4))

Now, we can use the power property to simplify the expression further:

log⁡(9x12(x3+4))=log⁡(9x12)+log⁡(x3+4)\log(9x^{\frac{1}{2}}(x^3+4)) = \log(9x^{\frac{1}{2}}) + \log(x^3+4)

Using the product property again, we can rewrite the expression as:

log⁡(9x12)+log⁡(x3+4)=log⁡(9x12(x3+4))\log(9x^{\frac{1}{2}}) + \log(x^3+4) = \log(9x^{\frac{1}{2}}(x^3+4))

Now, we can apply the quotient property to simplify the expression:

log⁡(9x12(x3+4))−log⁡6=log⁡(9x12(x3+4)6)\log(9x^{\frac{1}{2}}(x^3+4)) - \log 6 = \log\left(\frac{9x^{\frac{1}{2}}(x^3+4)}{6}\right)

Evaluating the Options

Now that we have simplified the given expression, let's evaluate the options:

A. log⁡3x(x3+4)2\log \frac{3 \sqrt{x}\left(x^3+4\right)}{2}

B. log⁡3x(3x+4)2\log \frac{3 \sqrt{x}(3 x+4)}{2}

C. log⁡3x(x3+4)2\log \frac{3 \sqrt{x}(x^3+4)}{2}

Comparing the simplified expression with the options, we can see that option C is the correct equivalent expression.

Conclusion

Simplifying logarithmic expressions requires a clear understanding of the properties of logarithms. By applying these properties, we can break down complex expressions into more manageable parts. In this article, we simplified a given logarithmic expression and identified the equivalent expression among the provided options. With practice and patience, you can master the art of simplifying logarithmic expressions and tackle even the most challenging problems.

Final Answer

The final answer is:

C\boxed{C}

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Introduction

Logarithmic expressions can be complex and challenging to simplify. However, with a clear understanding of the properties of logarithms, we can break down these expressions into more manageable parts. In this article, we will explore some common questions and answers related to logarithmic expressions.

Q&A

Q: What is the product property of logarithms?

A: The product property of logarithms states that log⁡(xy)=log⁡x+log⁡y\log(xy) = \log x + \log 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 simplify a logarithmic expression using the product property?

A: To simplify a logarithmic expression using the product property, you can break down the expression into individual factors and then apply the product property. For example, if you have the expression log⁡(3x2y)\log(3x^2y), you can break it down into log⁡3+log⁡x2+log⁡y\log 3 + \log x^2 + \log y.

Q: What is the quotient property of logarithms?

A: The quotient property of logarithms states that log⁡(xy)=log⁡x−log⁡y\log\left(\frac{x}{y}\right) = \log x - \log 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 simplify a logarithmic expression using the quotient property?

A: To simplify a logarithmic expression using the quotient property, you can break down the expression into individual factors and then apply the quotient property. For example, if you have the expression log⁡(xyz)\log\left(\frac{xy}{z}\right), you can break it down into log⁡x+log⁡y−log⁡z\log x + \log y - \log z.

Q: What is the power property of logarithms?

A: The power property of logarithms states that log⁡(xy)=ylog⁡x\log(x^y) = y\log 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 simplify a logarithmic expression using the power property?

A: To simplify a logarithmic expression using the power property, you can break down the expression into individual factors and then apply the power property. For example, if you have the expression log⁡(x3)\log(x^3), you can break it down into 3log⁡x3\log x.

Q: What is the change of base property of logarithms?

A: The change of base property of logarithms states that log⁡ba=log⁡calog⁡cb\log_b a = \frac{\log_c a}{\log_c b}. This means that the logarithm of a number with a different base can be expressed in terms of the logarithm of the same number with a different base.

Q: How do I apply the change of base property to simplify a logarithmic expression?

A: To apply the change of base property to simplify a logarithmic expression, you can express the logarithm of a number with a different base in terms of the logarithm of the same number with a different base. For example, if you have the expression log⁡53\log_5 3, you can express it as log⁡3log⁡5\frac{\log 3}{\log 5}.

Conclusion

Logarithmic expressions can be complex and challenging to simplify. However, with a clear understanding of the properties of logarithms, we can break down these expressions into more manageable parts. In this article, we explored some common questions and answers related to logarithmic expressions. With practice and patience, you can master the art of simplifying logarithmic expressions and tackle even the most challenging problems.

Final Tips

  • Always start by identifying the properties of logarithms that can be applied to the given expression.
  • Break down complex expressions into individual factors and then apply the properties of logarithms.
  • Use the change of base property to express logarithms with different bases in terms of a common base.
  • Practice, practice, practice! The more you practice simplifying logarithmic expressions, the more comfortable you will become with the properties of logarithms.