Expand The Logarithm:${ \log \frac{x^3}{z Y^4} }$A. ${ 3 \log (x) + \log (z) + \log (x) - 4 \log (z) - \log (x) + \log (z)\$} B. ${ 3 \log (x) - \log (z) - 4 \log (y)\$} C. ${ 3 \log (y)\$} D. ${ 4 \log (y)\$}

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Introduction

Logarithmic expressions are a fundamental concept in mathematics, and understanding how to expand them is crucial for solving various mathematical problems. In this article, we will focus on expanding the logarithm of a fraction, specifically the expression log⁡x3zy4\log \frac{x^3}{z y^4}. We will explore the properties of logarithms and apply them to simplify the given expression.

Properties of Logarithms

Before we dive into expanding the logarithmic expression, it's essential to understand the properties of logarithms. The logarithm of a product can be expressed as the sum of the logarithms of the individual terms. Mathematically, this can be represented as:

log⁡(a⋅b)=log⁡(a)+log⁡(b)\log (a \cdot b) = \log (a) + \log (b)

Similarly, the logarithm of a quotient can be expressed as the difference of the logarithms of the individual terms:

log⁡(ab)=log⁡(a)−log⁡(b)\log \left(\frac{a}{b}\right) = \log (a) - \log (b)

Expanding the Logarithmic Expression

Now that we have a solid understanding of the properties of logarithms, let's apply them to expand the given expression:

log⁡x3zy4\log \frac{x^3}{z y^4}

Using the property of logarithms for a quotient, we can rewrite the expression as:

log⁡(x3zy4)=log⁡(x3)−log⁡(zy4)\log \left(\frac{x^3}{z y^4}\right) = \log (x^3) - \log (z y^4)

Next, we can apply the property of logarithms for a product to expand the logarithm of the product zy4z y^4:

log⁡(x3)−log⁡(zy4)=log⁡(x3)−(log⁡(z)+log⁡(y4))\log (x^3) - \log (z y^4) = \log (x^3) - (\log (z) + \log (y^4))

Now, we can apply the property of logarithms for a power to expand the logarithm of x3x^3 and y4y^4:

log⁡(x3)−(log⁡(z)+log⁡(y4))=3log⁡(x)−log⁡(z)−4log⁡(y)\log (x^3) - (\log (z) + \log (y^4)) = 3 \log (x) - \log (z) - 4 \log (y)

Comparing the Expanded Expression with the Options

Now that we have expanded the logarithmic expression, let's compare it with the options provided:

A. 3log⁡(x)+log⁡(z)+log⁡(x)−4log⁡(z)−log⁡(x)+log⁡(z)3 \log (x) + \log (z) + \log (x) - 4 \log (z) - \log (x) + \log (z)

B. 3log⁡(x)−log⁡(z)−4log⁡(y)3 \log (x) - \log (z) - 4 \log (y)

C. 3log⁡(y)3 \log (y)

D. 4log⁡(y)4 \log (y)

Based on our expanded expression, we can see that option B is the correct answer.

Conclusion

In this article, we have explored the properties of logarithms and applied them to expand the logarithmic expression log⁡x3zy4\log \frac{x^3}{z y^4}. We have also compared the expanded expression with the options provided and identified the correct answer. Understanding how to expand logarithmic expressions is crucial for solving various mathematical problems, and we hope that this article has provided a clear and concise guide on how to do so.

Frequently Asked Questions

  • What is the property of logarithms for a quotient?
  • How do you expand the logarithm of a product?
  • What is the property of logarithms for a power?

Answers

  • The property of logarithms for a quotient is log⁡(ab)=log⁡(a)−log⁡(b)\log \left(\frac{a}{b}\right) = \log (a) - \log (b).
  • To expand the logarithm of a product, you can apply the property of logarithms for a product, which states that log⁡(a⋅b)=log⁡(a)+log⁡(b)\log (a \cdot b) = \log (a) + \log (b).
  • The property of logarithms for a power is log⁡(ab)=blog⁡(a)\log (a^b) = b \log (a).

Additional Resources

For more information on logarithmic expressions and their properties, we recommend checking out the following resources:

  • Khan Academy: Logarithms
  • Mathway: Logarithmic Expressions
  • Wolfram Alpha: Logarithmic Properties
    Logarithmic Expressions: A Q&A Guide =====================================

Introduction

Logarithmic expressions are a fundamental concept in mathematics, and understanding how to work with them is crucial for solving various mathematical problems. In this article, we will provide a Q&A guide on logarithmic expressions, covering topics such as properties, expansion, and simplification.

Q&A

Q: What is the property of logarithms for a quotient?

A: The property of logarithms for a quotient is log⁡(ab)=log⁡(a)−log⁡(b)\log \left(\frac{a}{b}\right) = \log (a) - \log (b).

Q: How do you expand the logarithm of a product?

A: To expand the logarithm of a product, you can apply the property of logarithms for a product, which states that log⁡(a⋅b)=log⁡(a)+log⁡(b)\log (a \cdot b) = \log (a) + \log (b).

Q: What is the property of logarithms for a power?

A: The property of logarithms for a power is log⁡(ab)=blog⁡(a)\log (a^b) = b \log (a).

Q: How do you simplify a logarithmic expression?

A: To simplify a logarithmic expression, you can apply the properties of logarithms, such as the quotient property, product property, and power property.

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

A: A logarithmic expression is the inverse of an exponential expression. For example, log⁡(x)=y\log (x) = y is equivalent to x=10yx = 10^y.

Q: How do you evaluate a logarithmic expression?

A: To evaluate a logarithmic expression, you can use a calculator or apply the properties of logarithms to simplify the expression.

Q: What is the base of a logarithmic expression?

A: The base of a logarithmic expression is the number that is used as the exponent in the exponential expression. For example, in the expression log⁡(x)\log (x), the base is 10.

Q: How do you change the base of a logarithmic expression?

A: To change the base of a logarithmic expression, you can use the change of base formula, which states that log⁡b(x)=log⁡a(x)log⁡a(b)\log_b (x) = \frac{\log_a (x)}{\log_a (b)}.

Q: What is the logarithmic identity?

A: The logarithmic identity is log⁡(a)+log⁡(b)=log⁡(a⋅b)\log (a) + \log (b) = \log (a \cdot b).

Q: How do you use logarithmic identities to simplify expressions?

A: To simplify expressions using logarithmic identities, you can apply the properties of logarithms, such as the quotient property, product property, and power property.

Common Mistakes

  • Forgetting to apply the properties of logarithms when simplifying expressions.
  • Not using the correct base when evaluating logarithmic expressions.
  • Not changing the base of a logarithmic expression when necessary.

Tips and Tricks

  • Use a calculator to evaluate logarithmic expressions when possible.
  • Apply the properties of logarithms to simplify expressions.
  • Change the base of a logarithmic expression when necessary.
  • Use logarithmic identities to simplify expressions.

Conclusion

In this article, we have provided a Q&A guide on logarithmic expressions, covering topics such as properties, expansion, and simplification. We hope that this guide has been helpful in understanding logarithmic expressions and how to work with them.