Expand The Logarithm Fully Using The Properties Of Logs. Express The Final Answer In Terms Of $\log Y$, $\log X$, And $\log Z$.$\log \frac{y 2}{\sqrt[3]{x 4} Z^2}$

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Introduction

Logarithms are a fundamental concept in mathematics, and understanding their properties is crucial for solving various mathematical problems. In this article, we will explore how to expand the logarithm of a given expression using the properties of logs. We will express the final answer in terms of $\log y$, $\log x$, and $\log z$.

The Properties of Logs

Before we dive into expanding the logarithm, let's review the properties of logs that we will use:

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

Expanding the Logarithm

Now, let's apply these properties to expand the given logarithm:

log⁑y2x43z2\log \frac{y^2}{\sqrt[3]{x^4} z^2}

We can start by applying the Quotient Rule:

log⁑y2x43z2=log⁑y2βˆ’log⁑x43z2\log \frac{y^2}{\sqrt[3]{x^4} z^2} = \log y^2 - \log \sqrt[3]{x^4} z^2

Next, we can apply the Power Rule to the first term:

log⁑y2=2log⁑y\log y^2 = 2 \log y

Now, let's focus on the second term. We can rewrite the cube root as a fractional exponent:

x43=x43\sqrt[3]{x^4} = x^{\frac{4}{3}}

So, the second term becomes:

log⁑x43z2=log⁑x43z2\log \sqrt[3]{x^4} z^2 = \log x^{\frac{4}{3}} z^2

We can now apply the Power Rule to this term:

log⁑x43z2=43log⁑x+2log⁑z\log x^{\frac{4}{3}} z^2 = \frac{4}{3} \log x + 2 \log z

Now, we can substitute this expression back into the original equation:

log⁑y2x43z2=2log⁑yβˆ’(43log⁑x+2log⁑z)\log \frac{y^2}{\sqrt[3]{x^4} z^2} = 2 \log y - \left(\frac{4}{3} \log x + 2 \log z\right)

Simplifying the Expression

We can simplify the expression by distributing the negative sign:

log⁑y2x43z2=2log⁑yβˆ’43log⁑xβˆ’2log⁑z\log \frac{y^2}{\sqrt[3]{x^4} z^2} = 2 \log y - \frac{4}{3} \log x - 2 \log z

Conclusion

In this article, we expanded the logarithm of a given expression using the properties of logs. We expressed the final answer in terms of $\log y$, $\log x$, and $\log z$. By applying the Product Rule, Quotient Rule, and Power Rule, we were able to simplify the expression and arrive at the final answer.

Key Takeaways

  • The Product Rule states that $\log (ab) = \log a + \log b$.
  • The Quotient Rule states that $\log \left(\frac{a}{b}\right) = \log a - \log b$.
  • The Power Rule states that $\log a^b = b \log a$.
  • We can expand logarithms using the properties of logs.
  • We can express the final answer in terms of $\log y$, $\log x$, and $\log z$.

Practice Problems

  1. Expand the logarithm: $\log \frac{x3}{y2 z^3}$
  2. Expand the logarithm: $\log \frac{y4}{\sqrt[5]{x5} z^4}$
  3. Expand the logarithm: $\log \frac{x^2 y3}{z4}$

Answer Key

  1. log⁑x3βˆ’log⁑y2βˆ’log⁑z3=3log⁑xβˆ’2log⁑yβˆ’3log⁑z\log x^3 - \log y^2 - \log z^3 = 3 \log x - 2 \log y - 3 \log z

  2. log⁑y4βˆ’log⁑x55z4=4log⁑yβˆ’55log⁑xβˆ’4log⁑z=4log⁑yβˆ’log⁑xβˆ’4log⁑z\log y^4 - \log \sqrt[5]{x^5} z^4 = 4 \log y - \frac{5}{5} \log x - 4 \log z = 4 \log y - \log x - 4 \log z

  3. \log x^2 y^3 - \log z^4 = 2 \log x + 3 \log y - 4 \log z$<br/>

Introduction

In our previous article, we explored how to expand logarithms using the properties of logs. We expressed the final answer in terms of $\log y$, $\log x$, and $\log z$. In this article, we will answer some frequently asked questions about logarithm expansion.

Q: What is the Product Rule for logarithms?

A: The Product Rule states that $\log (ab) = \log a + \log b$. This means that the logarithm of a product is equal to the sum of the logarithms of the individual factors.

Q: How do I apply the Quotient Rule for logarithms?

A: The Quotient Rule states that $\log \left(\frac{a}{b}\right) = \log a - \log b$. This means that the logarithm of a quotient is equal to the logarithm of the dividend minus the logarithm of the divisor.

Q: What is the Power Rule for logarithms?

A: The Power Rule states that $\log a^b = b \log a$. This means that the logarithm of a power is equal to the exponent multiplied by the logarithm of the base.

Q: How do I expand a logarithm with multiple terms?

A: To expand a logarithm with multiple terms, you can apply the Product Rule, Quotient Rule, and Power Rule in the following order:

  1. Apply the Quotient Rule to separate the logarithm into two terms.
  2. Apply the Power Rule to each term.
  3. Apply the Product Rule to combine the terms.

Q: Can I use logarithm expansion to simplify complex expressions?

A: Yes, logarithm expansion can be used to simplify complex expressions. By applying the properties of logs, you can break down complex expressions into simpler terms.

Q: What are some common mistakes to avoid when expanding logarithms?

A: Some common mistakes to avoid when expanding logarithms include:

  • Forgetting to apply the Quotient Rule when separating the logarithm into two terms.
  • Forgetting to apply the Power Rule when dealing with powers.
  • Not simplifying the expression after applying the properties of logs.

Q: How do I check my work when expanding logarithms?

A: To check your work when expanding logarithms, you can:

  • Verify that you have applied the correct properties of logs.
  • Simplify the expression to ensure that it is correct.
  • Check that the final answer is in the correct form.

Conclusion

In this article, we answered some frequently asked questions about logarithm expansion. We covered the Product Rule, Quotient Rule, and Power Rule, and provided tips for applying these rules to simplify complex expressions. By following these guidelines, you can become more confident in your ability to expand logarithms and simplify complex expressions.

Key Takeaways

  • The Product Rule states that $\log (ab) = \log a + \log b$.
  • The Quotient Rule states that $\log \left(\frac{a}{b}\right) = \log a - \log b$.
  • The Power Rule states that $\log a^b = b \log a$.
  • You can expand logarithms using the properties of logs.
  • You can simplify complex expressions using logarithm expansion.

Practice Problems

  1. Expand the logarithm: $\log \frac{x3}{y2 z^3}$
  2. Expand the logarithm: $\log \frac{y4}{\sqrt[5]{x5} z^4}$
  3. Expand the logarithm: $\log \frac{x^2 y3}{z4}$

Answer Key

  1. log⁑x3βˆ’log⁑y2βˆ’log⁑z3=3log⁑xβˆ’2log⁑yβˆ’3log⁑z\log x^3 - \log y^2 - \log z^3 = 3 \log x - 2 \log y - 3 \log z

  2. log⁑y4βˆ’log⁑x55z4=4log⁑yβˆ’55log⁑xβˆ’4log⁑z=4log⁑yβˆ’log⁑xβˆ’4log⁑z\log y^4 - \log \sqrt[5]{x^5} z^4 = 4 \log y - \frac{5}{5} \log x - 4 \log z = 4 \log y - \log x - 4 \log z

  3. log⁑x2y3βˆ’log⁑z4=2log⁑x+3log⁑yβˆ’4log⁑z\log x^2 y^3 - \log z^4 = 2 \log x + 3 \log y - 4 \log z