Choose The Property Used To Rewrite The Expression.\[$\log \sqrt[15]{125 X^3}=\frac{1}{5} \log 5 X\$\]A. Product Property B. Power Property C. Quotient Property D. Commutative Property

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Introduction

Logarithmic expressions can be rewritten using various properties to simplify them. In this article, we will focus on choosing the right property to rewrite the given expression: log⁡125x315=15log⁡5x\log \sqrt[15]{125 x^3}=\frac{1}{5} \log 5 x. We will explore the different properties of logarithms and provide examples to help you understand when to use each property.

Understanding Logarithmic Properties

Before we dive into the properties, let's recall the basic definition of a logarithm. A logarithm is the inverse operation of exponentiation. In other words, if y=axy = a^x, then log⁡ay=x\log_a y = x. With this in mind, let's explore the different properties of logarithms.

Product Property

The product property states that the logarithm of a product is equal to the sum of the logarithms of the individual terms. Mathematically, this can be expressed as:

log⁡(ab)=log⁡a+log⁡b\log (ab) = \log a + \log b

Power Property

The power property states that the logarithm of a power is equal to the exponent multiplied by the logarithm of the base. Mathematically, this can be expressed as:

log⁡ab=blog⁡a\log a^b = b \log a

Quotient Property

The quotient property states that the logarithm of a quotient is equal to the logarithm of the dividend minus the logarithm of the divisor. Mathematically, this can be expressed as:

log⁡ab=log⁡a−log⁡b\log \frac{a}{b} = \log a - \log b

Commutative Property

The commutative property states that the order of the terms does not change the result. Mathematically, this can be expressed as:

log⁡a+log⁡b=log⁡b+log⁡a\log a + \log b = \log b + \log a

Choosing the Right Property

Now that we have explored the different properties of logarithms, let's apply them to the given expression: log⁡125x315=15log⁡5x\log \sqrt[15]{125 x^3}=\frac{1}{5} \log 5 x. To rewrite this expression, we need to choose the right property.

Step 1: Simplify the Radicand

The first step is to simplify the radicand, which is 125x315\sqrt[15]{125 x^3}. We can rewrite this as:

125x315=53x315=5x\sqrt[15]{125 x^3} = \sqrt[15]{5^3 x^3} = 5 x

Step 2: Apply the Power Property

Now that we have simplified the radicand, we can apply the power property to rewrite the expression:

log⁡125x315=log⁡(5x)=log⁡5+log⁡x\log \sqrt[15]{125 x^3} = \log (5 x) = \log 5 + \log x

Step 3: Apply the Quotient Property

However, we are given that the expression is equal to 15log⁡5x\frac{1}{5} \log 5 x. To rewrite this expression, we need to apply the quotient property:

15log⁡5x=15(log⁡5+log⁡x)\frac{1}{5} \log 5 x = \frac{1}{5} (\log 5 + \log x)

Step 4: Simplify the Expression

Now that we have applied the quotient property, we can simplify the expression:

15(log⁡5+log⁡x)=15log⁡5+15log⁡x\frac{1}{5} (\log 5 + \log x) = \frac{1}{5} \log 5 + \frac{1}{5} \log x

Step 5: Apply the Power Property

Finally, we can apply the power property to rewrite the expression:

15log⁡5+15log⁡x=log⁡515+log⁡x15\frac{1}{5} \log 5 + \frac{1}{5} \log x = \log 5^{\frac{1}{5}} + \log x^{\frac{1}{5}}

Step 6: Simplify the Expression

Now that we have applied the power property, we can simplify the expression:

log⁡515+log⁡x15=log⁡55+log⁡x5\log 5^{\frac{1}{5}} + \log x^{\frac{1}{5}} = \log \sqrt[5]{5} + \log \sqrt[5]{x}

Step 7: Apply the Product Property

However, we are given that the expression is equal to log⁡125x315\log \sqrt[15]{125 x^3}. To rewrite this expression, we need to apply the product property:

log⁡125x315=log⁡55+log⁡x5\log \sqrt[15]{125 x^3} = \log \sqrt[5]{5} + \log \sqrt[5]{x}

Step 8: Simplify the Expression

Now that we have applied the product property, we can simplify the expression:

log⁡55+log⁡x5=log⁡5x5\log \sqrt[5]{5} + \log \sqrt[5]{x} = \log \sqrt[5]{5 x}

Conclusion

In conclusion, we have successfully rewritten the given expression using the product property. The final answer is:

log⁡125x315=log⁡5x5\log \sqrt[15]{125 x^3} = \log \sqrt[5]{5 x}

Answer

The correct answer is:

  • B. Power Property

Introduction

In our previous article, we explored the different properties of logarithms and applied them to rewrite a given expression. In this article, we will provide a Q&A section to help you better understand when to use each property.

Q: What is the Product Property?

A: The product property states that the logarithm of a product is equal to the sum of the logarithms of the individual terms. Mathematically, this can be expressed as:

log⁡(ab)=log⁡a+log⁡b\log (ab) = \log a + \log b

Q: When to use the Product Property?

A: You should use the product property when you have a product of two or more terms inside the logarithm. For example:

log⁡(2x)=log⁡2+log⁡x\log (2x) = \log 2 + \log x

Q: What is the Power Property?

A: The power property states that the logarithm of a power is equal to the exponent multiplied by the logarithm of the base. Mathematically, this can be expressed as:

log⁡ab=blog⁡a\log a^b = b \log a

Q: When to use the Power Property?

A: You should use the power property when you have a power inside the logarithm. For example:

log⁡x3=3log⁡x\log x^3 = 3 \log x

Q: What is the Quotient Property?

A: The quotient property states that the logarithm of a quotient is equal to the logarithm of the dividend minus the logarithm of the divisor. Mathematically, this can be expressed as:

log⁡ab=log⁡a−log⁡b\log \frac{a}{b} = \log a - \log b

Q: When to use the Quotient Property?

A: You should use the quotient property when you have a quotient inside the logarithm. For example:

log⁡xy=log⁡x−log⁡y\log \frac{x}{y} = \log x - \log y

Q: What is the Commutative Property?

A: The commutative property states that the order of the terms does not change the result. Mathematically, this can be expressed as:

log⁡a+log⁡b=log⁡b+log⁡a\log a + \log b = \log b + \log a

Q: When to use the Commutative Property?

A: You should use the commutative property when you have two or more logarithms that can be rearranged without changing the result. For example:

log⁡2+log⁡x=log⁡x+log⁡2\log 2 + \log x = \log x + \log 2

Q: How do I choose the right property to rewrite a logarithmic expression?

A: To choose the right property, you need to analyze the expression and identify the operations inside the logarithm. If you have a product, use the product property. If you have a power, use the power property. If you have a quotient, use the quotient property. If you have two or more logarithms that can be rearranged, use the commutative property.

Conclusion

In conclusion, choosing the right property to rewrite a logarithmic expression requires analyzing the expression and identifying the operations inside the logarithm. By understanding the different properties of logarithms and when to use each one, you can rewrite logarithmic expressions with ease.

Common Mistakes to Avoid

  • Not analyzing the expression carefully before choosing a property
  • Using the wrong property for the given expression
  • Not simplifying the expression after applying the property

Tips and Tricks

  • Practice, practice, practice! The more you practice rewriting logarithmic expressions, the more comfortable you will become with choosing the right property.
  • Use online resources, such as calculators or worksheets, to help you practice and reinforce your understanding.
  • Break down complex expressions into simpler ones by applying the properties of logarithms.

Final Thoughts

Rewriting logarithmic expressions is an essential skill in mathematics, and choosing the right property is crucial to simplifying the expression. By understanding the different properties of logarithms and when to use each one, you can rewrite logarithmic expressions with ease and confidence.