Use The Properties Of Logarithms To Expand $\log \frac{x^9}{z}$.Each Logarithm Should Involve Only One Variable And Should Not Have Any Exponents Or Fractions. Assume That All Variables Are Positive.

by ADMIN 202 views

Introduction

Logarithmic functions are a fundamental concept in mathematics, and they have numerous applications in various fields, including physics, engineering, and computer science. One of the key properties of logarithms is the ability to expand logarithmic expressions using various rules and formulas. In this article, we will explore how to use the properties of logarithms to expand the expression $\log \frac{x^9}{z}$, where each logarithm should involve only one variable and should not have any exponents or fractions.

The Quotient Rule

The quotient rule is one of the most important properties of logarithms, which states that:

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

This rule allows us to rewrite a logarithmic expression with a quotient as the argument as a difference of two logarithmic expressions.

Applying the Quotient Rule

Using the quotient rule, we can rewrite the given expression $\log \frac{x^9}{z}$ as:

log⁡x9z=log⁡x9−log⁡z\log \frac{x^9}{z} = \log x^9 - \log z

This is the first step in expanding the logarithmic expression.

The Power Rule

The power rule is another important property of logarithms, which states that:

log⁡ab=blog⁡a\log a^b = b \log a

This rule allows us to rewrite a logarithmic expression with an exponent as the argument as a multiple of the logarithmic expression without the exponent.

Applying the Power Rule

Using the power rule, we can rewrite the expression $\log x^9$ as:

log⁡x9=9log⁡x\log x^9 = 9 \log x

This is the second step in expanding the logarithmic expression.

Combining the Results

Now that we have applied both the quotient rule and the power rule, we can combine the results to get the final expanded form of the logarithmic expression:

log⁡x9z=9log⁡x−log⁡z\log \frac{x^9}{z} = 9 \log x - \log z

This is the final answer, and it meets the requirements of the problem, which states that each logarithm should involve only one variable and should not have any exponents or fractions.

Conclusion

In this article, we have used the properties of logarithms to expand the expression $\log \frac{x^9}{z}$. We have applied the quotient rule and the power rule to rewrite the expression as a difference of two logarithmic expressions, where each logarithm involves only one variable and does not have any exponents or fractions. This is a fundamental concept in mathematics, and it has numerous applications in various fields.

Example Problems

Here are some example problems that demonstrate the use of the properties of logarithms to expand logarithmic expressions:

  • log⁡y4w=log⁡y4−log⁡w=4log⁡y−log⁡w\log \frac{y^4}{w} = \log y^4 - \log w = 4 \log y - \log w

  • log⁡a2b3=log⁡a2−log⁡b3=2log⁡a−3log⁡b\log \frac{a^2}{b^3} = \log a^2 - \log b^3 = 2 \log a - 3 \log b

  • log⁡x5y2=log⁡x5−log⁡y2=5log⁡x−2log⁡y\log \frac{x^5}{y^2} = \log x^5 - \log y^2 = 5 \log x - 2 \log y

These example problems demonstrate the use of the quotient rule and the power rule to expand logarithmic expressions.

Practice Problems

Here are some practice problems that allow you to apply the properties of logarithms to expand logarithmic expressions:

  • log⁡z3x=log⁡z3−log⁡x=?\log \frac{z^3}{x} = \log z^3 - \log x = ?

  • log⁡a4b2=log⁡a4−log⁡b2=?\log \frac{a^4}{b^2} = \log a^4 - \log b^2 = ?

  • log⁡x2y3=log⁡x2−log⁡y3=?\log \frac{x^2}{y^3} = \log x^2 - \log y^3 = ?

These practice problems allow you to apply the quotient rule and the power rule to expand logarithmic expressions.

Solutions

Here are the solutions to the practice problems:

  • log⁡z3x=3log⁡z−log⁡x\log \frac{z^3}{x} = 3 \log z - \log x

  • log⁡a4b2=4log⁡a−2log⁡b\log \frac{a^4}{b^2} = 4 \log a - 2 \log b

  • log⁡x2y3=2log⁡x−3log⁡y\log \frac{x^2}{y^3} = 2 \log x - 3 \log y

These solutions demonstrate the use of the quotient rule and the power rule to expand logarithmic expressions.

Conclusion

Q: What is the quotient rule for logarithms?

A: The quotient rule for logarithms states that:

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

This rule allows us to rewrite a logarithmic expression with a quotient as the argument as a difference of two logarithmic expressions.

Q: How do I apply the quotient rule to expand a logarithmic expression?

A: To apply the quotient rule, simply rewrite the logarithmic expression with a quotient as the argument as a difference of two logarithmic expressions. For example:

log⁡x9z=log⁡x9−log⁡z\log \frac{x^9}{z} = \log x^9 - \log z

Q: What is the power rule for logarithms?

A: The power rule for logarithms states that:

log⁡ab=blog⁡a\log a^b = b \log a

This rule allows us to rewrite a logarithmic expression with an exponent as the argument as a multiple of the logarithmic expression without the exponent.

Q: How do I apply the power rule to expand a logarithmic expression?

A: To apply the power rule, simply rewrite the logarithmic expression with an exponent as the argument as a multiple of the logarithmic expression without the exponent. For example:

log⁡x9=9log⁡x\log x^9 = 9 \log x

Q: Can I use both the quotient rule and the power rule to expand a logarithmic expression?

A: Yes, you can use both the quotient rule and the power rule to expand a logarithmic expression. For example:

log⁡x9z=log⁡x9−log⁡z=9log⁡x−log⁡z\log \frac{x^9}{z} = \log x^9 - \log z = 9 \log x - \log z

Q: What are some common logarithmic expressions that can be expanded using the quotient rule and the power rule?

A: Some common logarithmic expressions that can be expanded using the quotient rule and the power rule include:

  • log⁡y4w=log⁡y4−log⁡w=4log⁡y−log⁡w\log \frac{y^4}{w} = \log y^4 - \log w = 4 \log y - \log w

  • log⁡a2b3=log⁡a2−log⁡b3=2log⁡a−3log⁡b\log \frac{a^2}{b^3} = \log a^2 - \log b^3 = 2 \log a - 3 \log b

  • log⁡x5y2=log⁡x5−log⁡y2=5log⁡x−2log⁡y\log \frac{x^5}{y^2} = \log x^5 - \log y^2 = 5 \log x - 2 \log y

Q: How do I know which rule to apply first when expanding a logarithmic expression?

A: When expanding a logarithmic expression, it's usually best to apply the quotient rule first, followed by the power rule. This will help you to simplify the expression and make it easier to work with.

Q: Can I use logarithmic properties to simplify expressions with multiple logarithms?

A: Yes, you can use logarithmic properties to simplify expressions with multiple logarithms. For example:

log⁡(x2y3)=log⁡x2+log⁡y3=2log⁡x+3log⁡y\log (x^2y^3) = \log x^2 + \log y^3 = 2 \log x + 3 \log y

Q: What are some real-world applications of logarithmic properties?

A: Logarithmic properties have numerous real-world applications, including:

  • Physics: Logarithmic properties are used to describe the behavior of physical systems, such as the decay of radioactive materials.
  • Engineering: Logarithmic properties are used to design and analyze complex systems, such as electronic circuits and mechanical systems.
  • Computer Science: Logarithmic properties are used to develop algorithms and data structures, such as binary search trees and hash tables.

Conclusion

In this article, we have answered some common questions about logarithmic properties and how to apply them to expand logarithmic expressions. We have also discussed some common logarithmic expressions that can be expanded using the quotient rule and the power rule, as well as some real-world applications of logarithmic properties.