Evaluate The Expression $\log _5 \frac{1}{25}$.A. $-\frac{1}{2}$ B. $-2$ C. $\frac{1}{2}$ D. $2$

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Introduction

In this article, we will delve into the world of logarithms and evaluate the expression $\log _5 \frac{1}{25}$. Logarithms are a fundamental concept in mathematics, and understanding how to evaluate them is crucial for solving various mathematical problems. The expression $\log _5 \frac{1}{25}$ involves a logarithm with base 5 and an argument of $\frac{1}{25}$. Our goal is to simplify this expression and determine its value.

Understanding Logarithms

Before we proceed with evaluating the expression, let's briefly review the concept of logarithms. A logarithm is the inverse operation of exponentiation. In other words, if we have an equation of the form $a^x = b$, then the logarithm of $b$ with base $a$ is equal to $x$. This can be expressed mathematically as:

$$\log_a b = x \iff a^x = b$$

Evaluating the Expression

Now that we have a basic understanding of logarithms, let's focus on evaluating the expression $\log _5 \frac{1}{25}$. To do this, we can start by rewriting the argument of the logarithm in a more convenient form. We know that $\frac{1}{25}$ can be expressed as $5^{-2}$, since $5^{-2} = \frac{1}{5^2} = \frac{1}{25}$.

Using this information, we can rewrite the expression as:

$$\log _5 \frac{1}{25} = \log _5 5^{-2}$$

Applying Logarithmic Properties

Now that we have rewritten the expression, we can apply some logarithmic properties to simplify it further. One of the key properties of logarithms is that the logarithm of a power is equal to the exponent multiplied by the logarithm of the base. Mathematically, this can be expressed as:

$$\log_a b^x = x \log_a b$$

Using this property, we can simplify the expression as follows:

$$\log _5 5^{-2} = -2 \log _5 5$$

Evaluating the Logarithm

Now that we have simplified the expression, we can evaluate the logarithm. We know that the logarithm of a number with the same base is equal to 1. In this case, we have:

$$\log _5 5 = 1$$

Therefore, we can substitute this value into the expression:

$$-2 \log _5 5 = -2 \cdot 1 = -2$$

Conclusion

In conclusion, we have evaluated the expression $\log _5 \frac{1}{25}$ and determined its value to be $-2$. This result is consistent with the properties of logarithms and demonstrates the importance of understanding these concepts in mathematics.

Final Answer

The final answer to the expression $\log _5 \frac{1}{25}$ is:

• A. $-\frac{1}{2}$: Incorrect
• B. $-2$: Correct
• C. $\frac{1}{2}$: Incorrect
• D. $2$: Incorrect

The correct answer is B. $-2$.

Introduction

In our previous article, we evaluated the expression $\log _5 \frac{1}{25}$ and determined its value to be $-2$. Logarithmic expressions can be complex and challenging to evaluate, but with a solid understanding of the underlying concepts and properties, you can tackle even the most difficult problems. In this article, we will provide a Q&A guide to help you better understand logarithmic expressions and how to evaluate them.

Q1: What is the definition of a logarithm?

A1: A logarithm is the inverse operation of exponentiation. In other words, if we have an equation of the form $a^x = b$, then the logarithm of $b$ with base $a$ is equal to $x$. This can be expressed mathematically as:

$$\log_a b = x \iff a^x = b$$

Q2: How do I evaluate a logarithmic expression?

A2: To evaluate a logarithmic expression, you need to follow these steps:

1. Rewrite the argument of the logarithm in a more convenient form.
2. Apply logarithmic properties to simplify the expression.
3. Evaluate the logarithm using the properties of logarithms.

Q3: What is the property of logarithms that states the logarithm of a power is equal to the exponent multiplied by the logarithm of the base?

A3: The property of logarithms that states the logarithm of a power is equal to the exponent multiplied by the logarithm of the base is:

$$\log_a b^x = x \log_a b$$

Q4: How do I apply the property of logarithms to simplify an expression?

A4: To apply the property of logarithms to simplify an expression, you need to follow these steps:

1. Identify the power and the base in the expression.
2. Rewrite the expression using the property of logarithms.
3. Simplify the expression by evaluating the logarithm.

Q5: What is the value of $\log _5 5$?

A5: The value of $\log _5 5$ is 1, since the logarithm of a number with the same base is equal to 1.

Q6: How do I evaluate the expression $\log _5 \frac{1}{25}$?

A6: To evaluate the expression $\log _5 \frac{1}{25}$, you need to follow these steps:

1. Rewrite the argument of the logarithm in a more convenient form.
2. Apply logarithmic properties to simplify the expression.
3. Evaluate the logarithm using the properties of logarithms.

Using these steps, we can evaluate the expression as follows:

$$\log _5 \frac{1}{25} = \log _5 5^{-2} = -2 \log _5 5 = -2 \cdot 1 = -2$$

Q7: What is the final answer to the expression $\log _5 \frac{1}{25}$?

A7: The final answer to the expression $\log _5 \frac{1}{25}$ is $-2$.

Q8: What are some common mistakes to avoid when evaluating logarithmic expressions?

A8: Some common mistakes to avoid when evaluating logarithmic expressions include:

• Not rewriting the argument of the logarithm in a more convenient form.
• Not applying logarithmic properties to simplify the expression.
• Not evaluating the logarithm using the properties of logarithms.

Q9: How can I practice evaluating logarithmic expressions?

A9: You can practice evaluating logarithmic expressions by working through examples and exercises. You can also use online resources and calculators to help you evaluate logarithmic expressions.

Q10: What are some real-world applications of logarithmic expressions?

A10: Logarithmic expressions have many real-world applications, including:

• Calculating the pH of a solution.
• Determining the magnitude of an earthquake.
• Analyzing the growth of a population.

By understanding logarithmic expressions and how to evaluate them, you can tackle a wide range of mathematical problems and apply your knowledge to real-world situations.

Final Answer

The final answer to the expression $\log _5 \frac{1}{25}$ is:

• A. $-\frac{1}{2}$: Incorrect
• B. $-2$: Correct
• C. $\frac{1}{2}$: Incorrect
• D. $2$: Incorrect

The correct answer is B. $-2$.