Using Numerical Analysis And The Power Rule Of Logarithms, $\log X^k = K \log X$, How Can $\log \left(\frac{1}{100}\right$\] Be Rewritten?A. $2 \log (-10$\]B. $10 \log (-2$\]C. $-2 \log 10$D. $\log (-20$\]

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Introduction

In mathematics, logarithmic expressions are a crucial part of various mathematical operations. The power rule of logarithms, which states that log⁡xk=klog⁡x\log x^k = k \log x, is a fundamental concept in rewriting logarithmic expressions. In this article, we will explore how to use numerical analysis and the power rule of logarithms to rewrite the expression log⁡(1100)\log \left(\frac{1}{100}\right).

Understanding the Power Rule of Logarithms

The power rule of logarithms is a fundamental concept in mathematics that allows us to rewrite logarithmic expressions in a simpler form. The power rule states that log⁡xk=klog⁡x\log x^k = k \log x. This means that we can rewrite a logarithmic expression with a power as a product of the power and the logarithm of the base.

Applying the Power Rule to the Given Expression

To rewrite the expression log⁡(1100)\log \left(\frac{1}{100}\right), we can use the power rule of logarithms. We can rewrite 1100\frac{1}{100} as 10−210^{-2}, since 10−2=110010^{-2} = \frac{1}{100}. Now, we can apply the power rule to rewrite the expression as:

log⁡(1100)=log⁡(10−2)\log \left(\frac{1}{100}\right) = \log (10^{-2})

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

log⁡(10−2)=−2log⁡10\log (10^{-2}) = -2 \log 10

Evaluating the Options

Now that we have rewritten the expression using the power rule, we can evaluate the options:

A. 2log⁡(−10)2 \log (-10) B. 10log⁡(−2)10 \log (-2) C. −2log⁡10-2 \log 10 D. log⁡(−20)\log (-20)

Based on our rewritten expression, we can see that option C is the correct answer.

Conclusion

In conclusion, we have used numerical analysis and the power rule of logarithms to rewrite the expression log⁡(1100)\log \left(\frac{1}{100}\right). We have shown that the power rule allows us to rewrite logarithmic expressions in a simpler form, and we have evaluated the options to determine the correct answer.

Key Takeaways

  • The power rule of logarithms states that log⁡xk=klog⁡x\log x^k = k \log x.
  • We can rewrite a logarithmic expression with a power as a product of the power and the logarithm of the base.
  • We can use numerical analysis and the power rule to rewrite logarithmic expressions in a simpler form.

Additional Examples

Here are some additional examples of using the power rule of logarithms:

  • log⁡(x3)=3log⁡x\log (x^3) = 3 \log x
  • log⁡(y−4)=−4log⁡y\log (y^{-4}) = -4 \log y
  • log⁡(z2)=2log⁡z\log (z^2) = 2 \log z

These examples demonstrate how the power rule can be used to rewrite logarithmic expressions in a simpler form.

Common Mistakes

When using the power rule of logarithms, it is common to make mistakes such as:

  • Forgetting to apply the power rule
  • Applying the power rule incorrectly
  • Not simplifying the expression after applying the power rule

To avoid these mistakes, it is essential to carefully read and understand the power rule, and to apply it correctly.

Real-World Applications

The power rule of logarithms has numerous real-world applications in various fields, including:

  • Engineering: The power rule is used to calculate the logarithm of a quantity with a power, which is essential in engineering applications such as signal processing and control systems.
  • Computer Science: The power rule is used in computer science to calculate the logarithm of a quantity with a power, which is essential in algorithms and data structures.
  • Economics: The power rule is used in economics to calculate the logarithm of a quantity with a power, which is essential in econometrics and economic modeling.

Q: What is the power rule of logarithms?

A: The power rule of logarithms states that log⁡xk=klog⁡x\log x^k = k \log x. This means that we can rewrite a logarithmic expression with a power as a product of the power and the logarithm of the base.

Q: How do I apply the power rule of logarithms?

A: To apply the power rule, simply rewrite the logarithmic expression with a power as a product of the power and the logarithm of the base. For example, log⁡(x3)=3log⁡x\log (x^3) = 3 \log x.

Q: What are some common mistakes to avoid when using the power rule of logarithms?

A: Some common mistakes to avoid when using the power rule of logarithms include:

  • Forgetting to apply the power rule
  • Applying the power rule incorrectly
  • Not simplifying the expression after applying the power rule

Q: How do I simplify a logarithmic expression after applying the power rule?

A: To simplify a logarithmic expression after applying the power rule, simply multiply the power by the logarithm of the base. For example, log⁡(x3)=3log⁡x\log (x^3) = 3 \log x can be simplified to 3log⁡x3 \log x.

Q: Can I use the power rule of logarithms with negative powers?

A: Yes, you can use the power rule of logarithms with negative powers. For example, log⁡(x−3)=−3log⁡x\log (x^{-3}) = -3 \log x.

Q: Can I use the power rule of logarithms with fractional powers?

A: Yes, you can use the power rule of logarithms with fractional powers. For example, log⁡(x1/2)=12log⁡x\log (x^{1/2}) = \frac{1}{2} \log x.

Q: How do I evaluate a logarithmic expression with a power using the power rule?

A: To evaluate a logarithmic expression with a power using the power rule, simply apply the power rule and then simplify the expression. For example, log⁡(x3)=3log⁡x\log (x^3) = 3 \log x can be evaluated as 3log⁡x3 \log x.

Q: Can I use the power rule of logarithms with logarithmic expressions inside other logarithmic expressions?

A: Yes, you can use the power rule of logarithms with logarithmic expressions inside other logarithmic expressions. For example, log⁡(log⁡x3)=log⁡(3log⁡x)\log (\log x^3) = \log (3 \log x).

Q: How do I apply the power rule of logarithms to logarithmic expressions with multiple powers?

A: To apply the power rule of logarithms to logarithmic expressions with multiple powers, simply apply the power rule to each power separately. For example, log⁡(x3y2)=3log⁡x+2log⁡y\log (x^3 y^2) = 3 \log x + 2 \log y.

Q: Can I use the power rule of logarithms with logarithmic expressions that have a base other than 10?

A: Yes, you can use the power rule of logarithms with logarithmic expressions that have a base other than 10. For example, log⁡b(x3)=3log⁡bx\log_b (x^3) = 3 \log_b x.

Conclusion

In conclusion, the power rule of logarithms is a fundamental concept in mathematics that allows us to rewrite logarithmic expressions in a simpler form. By understanding and applying the power rule, we can solve a wide range of mathematical problems and real-world applications.