Using Numerical Analysis And The Power Rule Of Logarithms, How Can \log \left(\frac{1}{196}\right ] Be Rewritten?A. 14 Log ⁡ − 2 14 \log -2 14 Lo G − 2 B. 2 Log ⁡ − 14 2 \log -14 2 Lo G − 14 C. − 2 Log ⁡ 14 -2 \log 14 − 2 Lo G 14 D. Log ⁡ − 28 \log -28 Lo G − 28 Check Your Answer.

Introduction

In mathematics, logarithmic expressions are a fundamental concept in algebra and calculus. The power rule of logarithms is a crucial tool for rewriting and simplifying logarithmic expressions. In this article, we will explore how to use numerical analysis and the power rule of logarithms to rewrite the expression $\log \left(\frac{1}{196}\right)$.

Understanding the Power Rule of Logarithms

The power rule of logarithms states that for any positive real numbers $a$ and $b$, and any real number $c$:

$$\log_a (b^c) = c \log_a b$$

This rule allows us to rewrite logarithmic expressions by moving the exponent from the inside of the logarithm to the outside.

Rewriting the Expression

To rewrite the expression $\log \left(\frac{1}{196}\right)$, we can start by analyzing the fraction inside the logarithm. We can rewrite $\frac{1}{196}$ as $196^{-1}$, which is equivalent to $(14^2)^{-1}$.

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

$$\log \left(\frac{1}{196}\right) = \log (196^{-1}) = -1 \log 196$$

However, we can further simplify this expression by using the fact that $196 = 14^2$. Therefore, we can rewrite the expression as:

$$\log \left(\frac{1}{196}\right) = -1 \log (14^2)$$

Now, we can apply the power rule of logarithms again to rewrite the expression as:

$$\log \left(\frac{1}{196}\right) = -2 \log 14$$

Checking the Answer

To check our answer, we can plug in the value of $14$ into the expression $-2 \log 14$. Using a calculator or a logarithmic table, we can find that:

$$-2 \log 14 = -2 \cdot 1.1461 = -2.2922$$

However, we can also rewrite the expression $\log \left(\frac{1}{196}\right)$ as:

$$\log \left(\frac{1}{196}\right) = \log (196^{-1}) = \log \left(\frac{1}{14^2}\right)$$

Using the fact that $\log \left(\frac{1}{x}\right) = -\log x$, we can rewrite the expression as:

$$\log \left(\frac{1}{196}\right) = -\log (14^2)$$

Now, we can apply the power rule of logarithms again to rewrite the expression as:

$$\log \left(\frac{1}{196}\right) = -2 \log 14$$

This confirms that our original answer, $-2 \log 14$, is correct.

Conclusion

In conclusion, we have used numerical analysis and the power rule of logarithms to rewrite the expression $\log \left(\frac{1}{196}\right)$. By analyzing the fraction inside the logarithm and applying the power rule of logarithms, we were able to rewrite the expression as $-2 \log 14$. This confirms that the correct answer is indeed $-2 \log 14$.

Final Answer

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Introduction

In our previous article, we explored how to use numerical analysis and the power rule of logarithms to rewrite the expression $\log \left(\frac{1}{196}\right)$. In this article, we will answer some frequently asked questions about the power rule of logarithms.

Q: What is the power rule of logarithms?

A: The power rule of logarithms states that for any positive real numbers $a$ and $b$, and any real number $c$:

$$\log_a (b^c) = c \log_a b$$

This rule allows us to rewrite logarithmic expressions by moving the exponent from the inside of the logarithm to the outside.

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

A: To apply the power rule of logarithms, you need to identify the exponent inside the logarithm and move it to the outside. For example, if you have the expression $\log (x^2)$, you can rewrite it as $2 \log x$ using the power rule of logarithms.

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 change the sign of the exponent when moving it to the outside of the logarithm.
• Not checking if the base of the logarithm is positive.
• Not simplifying the expression after applying the power rule of logarithms.

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

A: Yes, you can use the power rule of logarithms with negative exponents. For example, if you have the expression $\log (x^{-2})$, you can rewrite it as $-2 \log x$ using the power rule of logarithms.

Q: How do I check my answer when using the power rule of logarithms?

A: To check your answer when using the power rule of logarithms, you can plug in the value of the variable into the expression and simplify. For example, if you have the expression $2 \log x$ and you plug in $x = 4$, you can rewrite it as $2 \log 4$ and simplify to get $2 \cdot 0.6021 = 1.2042$.

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, if you have the expression $\log (\log x)$, you can rewrite it as $\log (x^{\log x})$ using the power rule of logarithms.

Conclusion

In conclusion, we have answered some frequently asked questions about the power rule of logarithms. By understanding the power rule of logarithms and how to apply it, you can simplify complex logarithmic expressions and solve problems more efficiently.

Final Tips

• Always check your answer when using the power rule of logarithms.
• Simplify the expression after applying the power rule of logarithms.
• Use the power rule of logarithms with negative exponents.
• Use the power rule of logarithms with logarithmic expressions inside other logarithmic expressions.

Common Mistakes to Avoid

• Forgetting to change the sign of the exponent when moving it to the outside of the logarithm.
• Not checking if the base of the logarithm is positive.
• Not simplifying the expression after applying the power rule of logarithms.

Power Rule of Logarithms Formula

$$\log_a (b^c) = c \log_a b$$

Power Rule of Logarithms Examples

• $\log (x^2) = 2 \log x$
• $\log (x^{-2}) = -2 \log x$
• $\log (\log x) = \log (x^{\log x})$

Power Rule of Logarithms Practice Problems

• $\log (x^3) = ?$
• $\log (x^{-4}) = ?$
• $\log (\log x) = ?$

Answer Key

• $\log (x^3) = 3 \log x$
• $\log (x^{-4}) = -4 \log x$
• $\log (\log x) = \log (x^{\log x})$