Evaluate The Expression $\log_5 \frac{1}{25}$.A. $-\frac{1}{2}$ B. $-2$ C. $\frac{1}{2}$ D. $2$

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Understanding the Problem

The given problem involves evaluating the logarithmic expression $\log_5 \frac{1}{25}$. To solve this, we need to understand the properties of logarithms and how to apply them to simplify the expression.

Properties of Logarithms

A logarithm is the inverse operation of exponentiation. In other words, if $x = a^y$, then $y = \log_a x$. This means that the logarithm of a number is the exponent to which the base must be raised to produce that number.

One of the key properties of logarithms is the change of base formula, which states that $\log_b a = \frac{\log_c a}{\log_c b}$ for any positive numbers $a, b,$ and $c$ where $c \neq 1$. This formula allows us to change the base of a logarithm to any other base.

Simplifying the Expression

To simplify the expression $\log_5 \frac{1}{25}$, we can start by rewriting the fraction as a product of powers of 5. Since $25 = 5^2$, we can write $\frac{1}{25} = \frac{1}{5^2} = 5^{-2}$.

Applying the Property of Negative Exponents

Using the property of negative exponents, we know that $a^{-n} = \frac{1}{a^n}$. Therefore, we can rewrite $5^{-2}$ as $\frac{1}{5^2}$.

Evaluating the Logarithm

Now that we have simplified the expression to $\log_5 \frac{1}{25} = \log_5 5^{-2}$, we can evaluate the logarithm. Since the base and the argument of the logarithm are the same, we know that $\log_5 5^{-2} = -2$.

Conclusion

In conclusion, the value of the expression $\log_5 \frac{1}{25}$ is $-2$. This is because we can simplify the expression by rewriting the fraction as a product of powers of 5, and then applying the property of negative exponents.

Final Answer

The final answer is B. $-2$.

Step-by-Step Solution

Here's a step-by-step solution to the problem:

1. Rewrite the fraction as a product of powers of 5: $\frac{1}{25} = \frac{1}{5^2} = 5^{-2}$.
2. Apply the property of negative exponents: $5^{-2} = \frac{1}{5^2}$.
3. Evaluate the logarithm: $\log_5 5^{-2} = -2$.

Common Mistakes

Here are some common mistakes to avoid when solving this problem:

• Not rewriting the fraction as a product of powers of 5.
• Not applying the property of negative exponents.
• Not evaluating the logarithm correctly.

Tips and Tricks

Here are some tips and tricks to help you solve this problem:

• Make sure to rewrite the fraction as a product of powers of 5.
• Apply the property of negative exponents to simplify the expression.
• Evaluate the logarithm correctly by using the property of logarithms.

Real-World Applications

This problem has real-world applications in fields such as engineering, physics, and computer science. For example, logarithms are used to calculate the decibel level of a sound, the pH level of a solution, and the magnitude of an earthquake.

Conclusion

In conclusion, the value of the expression $\log_5 \frac{1}{25}$ is $-2$. This is because we can simplify the expression by rewriting the fraction as a product of powers of 5, and then applying the property of negative exponents.

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Frequently Asked Questions

Q: What is the value of the expression $\log_5 \frac{1}{25}$?

A: The value of the expression $\log_5 \frac{1}{25}$ is $-2$.

Q: How do I simplify the expression $\log_5 \frac{1}{25}$?

A: To simplify the expression $\log_5 \frac{1}{25}$, you can rewrite the fraction as a product of powers of 5. Since $25 = 5^2$, you can write $\frac{1}{25} = \frac{1}{5^2} = 5^{-2}$.

Q: What is the property of negative exponents that I need to apply?

A: The property of negative exponents states that $a^{-n} = \frac{1}{a^n}$. Therefore, you can rewrite $5^{-2}$ as $\frac{1}{5^2}$.

Q: How do I evaluate the logarithm $\log_5 5^{-2}$?

A: Since the base and the argument of the logarithm are the same, you know that $\log_5 5^{-2} = -2$.

Q: What are some common mistakes to avoid when solving this problem?

A: Some common mistakes to avoid when solving this problem include:

• Not rewriting the fraction as a product of powers of 5.
• Not applying the property of negative exponents.
• Not evaluating the logarithm correctly.

Q: What are some tips and tricks to help me solve this problem?

A: Some tips and tricks to help you solve this problem include:

• Make sure to rewrite the fraction as a product of powers of 5.
• Apply the property of negative exponents to simplify the expression.
• Evaluate the logarithm correctly by using the property of logarithms.

Q: What are some real-world applications of logarithms?

A: Logarithms have real-world applications in fields such as engineering, physics, and computer science. For example, logarithms are used to calculate the decibel level of a sound, the pH level of a solution, and the magnitude of an earthquake.

Q: Can I use a calculator to evaluate the logarithm?

A: Yes, you can use a calculator to evaluate the logarithm. However, it's also important to understand the underlying math and be able to solve the problem without a calculator.

Q: What is the change of base formula for logarithms?

A: The change of base formula for logarithms states that $\log_b a = \frac{\log_c a}{\log_c b}$ for any positive numbers $a, b,$ and $c$ where $c \neq 1$. This formula allows you to change the base of a logarithm to any other base.

Q: Can I use the change of base formula to evaluate the logarithm?

A: Yes, you can use the change of base formula to evaluate the logarithm. However, in this case, it's not necessary since the base and the argument of the logarithm are the same.

Q: What is the final answer to the problem?

A: The final answer to the problem is $-2$.

Conclusion

In conclusion, the value of the expression $\log_5 \frac{1}{25}$ is $-2$. This is because we can simplify the expression by rewriting the fraction as a product of powers of 5, and then applying the property of negative exponents. We also discussed some common mistakes to avoid, tips and tricks to help you solve the problem, and real-world applications of logarithms.