The Expression 1 3 Log ⁡ M − 2 Log ⁡ N \frac{1}{3} \log M - 2 \log N 3 1 ​ Lo G M − 2 Lo G N Is Equivalent To:A. \log \left(\frac{1}{3} M - 2 N\right ]B. \log \left(\frac{m^3}{\sqrt{n}}\right ]C. \log \left(\sqrt[3]{m} - N^2\right ]D. $\log

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Understanding the Problem

The given expression involves logarithmic functions and requires us to simplify it to one of the provided options. To begin with, we need to understand the properties of logarithms and how they can be manipulated to simplify the given expression.

Properties of Logarithms

Before diving into the solution, let's recall some essential properties of logarithms:

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

Simplifying the Expression

Now, let's focus on simplifying the given expression using the properties of logarithms.

13log⁡m−2log⁡n\frac{1}{3} \log m - 2 \log n

We can rewrite the expression as:

log⁡m13−log⁡n2\log m^{\frac{1}{3}} - \log n^2

Using the power rule, we can rewrite the expression as:

log⁡(m13n2)\log \left(\frac{m^{\frac{1}{3}}}{n^2}\right)

Evaluating the Options

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

A. log⁡(13m−2n)\log \left(\frac{1}{3} m - 2 n\right)

This option does not match our simplified expression.

B. log⁡(m3n)\log \left(\frac{m^3}{\sqrt{n}}\right)

We can rewrite this option as:

log⁡(m3n12)\log \left(\frac{m^3}{n^{\frac{1}{2}}}\right)

Using the quotient rule, we can rewrite this option as:

log⁡m3−log⁡n12\log m^3 - \log n^{\frac{1}{2}}

Using the power rule, we can rewrite this option as:

3log⁡m−12log⁡n3 \log m - \frac{1}{2} \log n

This option does not match our simplified expression.

C. log⁡(m3−n2)\log \left(\sqrt[3]{m} - n^2\right)

This option does not match our simplified expression.

D. log⁡(m13n2)\log \left(\frac{m^{\frac{1}{3}}}{n^2}\right)

This option matches our simplified expression.

Conclusion

In conclusion, the expression 13log⁡m−2log⁡n\frac{1}{3} \log m - 2 \log n is equivalent to log⁡(m13n2)\log \left(\frac{m^{\frac{1}{3}}}{n^2}\right).

Final Answer

The final answer is D\boxed{D}.

Explanation

The expression 13log⁡m−2log⁡n\frac{1}{3} \log m - 2 \log n can be simplified using the properties of logarithms. By applying the power rule, we can rewrite the expression as log⁡(m13n2)\log \left(\frac{m^{\frac{1}{3}}}{n^2}\right). This matches option D, which is the correct answer.

Key Takeaways

  • The expression 13log⁡m−2log⁡n\frac{1}{3} \log m - 2 \log n can be simplified using the properties of logarithms.
  • The power rule can be used to rewrite the expression as log⁡(m13n2)\log \left(\frac{m^{\frac{1}{3}}}{n^2}\right).
  • The correct answer is option D, which matches the simplified expression.

Common Mistakes

  • Not applying the power rule correctly.
  • Not using the quotient rule to simplify the expression.
  • Not rewriting the expression in the correct form.

Tips and Tricks

  • Make sure to apply the power rule correctly.
  • Use the quotient rule to simplify the expression.
  • Rewrite the expression in the correct form.

Real-World Applications

  • The expression 13log⁡m−2log⁡n\frac{1}{3} \log m - 2 \log n can be used in real-world applications such as finance and engineering.
  • The properties of logarithms can be used to simplify complex expressions and make them easier to work with.

Conclusion

In conclusion, the expression 13log⁡m−2log⁡n\frac{1}{3} \log m - 2 \log n is equivalent to log⁡(m13n2)\log \left(\frac{m^{\frac{1}{3}}}{n^2}\right). This can be simplified using the properties of logarithms, and the correct answer is option D.

Frequently Asked Questions

Q: What is the expression 13log⁡m−2log⁡n\frac{1}{3} \log m - 2 \log n equivalent to?

A: The expression 13log⁡m−2log⁡n\frac{1}{3} \log m - 2 \log n is equivalent to log⁡(m13n2)\log \left(\frac{m^{\frac{1}{3}}}{n^2}\right).

Q: How do I simplify the expression 13log⁡m−2log⁡n\frac{1}{3} \log m - 2 \log n?

A: To simplify the expression, you can use the properties of logarithms. Specifically, you can apply the power rule to rewrite the expression as log⁡(m13n2)\log \left(\frac{m^{\frac{1}{3}}}{n^2}\right).

Q: What is the power rule of logarithms?

A: The power rule of logarithms states that log⁡(ab)=blog⁡a\log (a^b) = b \log a. This can be used to rewrite the expression 13log⁡m−2log⁡n\frac{1}{3} \log m - 2 \log n as log⁡(m13n2)\log \left(\frac{m^{\frac{1}{3}}}{n^2}\right).

Q: How do I use the quotient rule to simplify the expression?

A: To use the quotient rule, you can rewrite the expression as log⁡(m13n2)\log \left(\frac{m^{\frac{1}{3}}}{n^2}\right). This can be simplified further by applying the power rule.

Q: What is the correct answer?

A: The correct answer is option D, which is log⁡(m13n2)\log \left(\frac{m^{\frac{1}{3}}}{n^2}\right).

Q: What are some common mistakes to avoid when simplifying the expression?

A: Some common mistakes to avoid include not applying the power rule correctly, not using the quotient rule to simplify the expression, and not rewriting the expression in the correct form.

Q: What are some tips and tricks for simplifying the expression?

A: Some tips and tricks for simplifying the expression include making sure to apply the power rule correctly, using the quotient rule to simplify the expression, and rewriting the expression in the correct form.

Q: How can I apply the expression in real-world scenarios?

A: The expression 13log⁡m−2log⁡n\frac{1}{3} \log m - 2 \log n can be used in real-world scenarios such as finance and engineering. The properties of logarithms can be used to simplify complex expressions and make them easier to work with.

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