Which Of The Following Is Equivalent To Log ⁡ 1 9 K \log \frac{\frac{1}{9}}{k} Lo G K 9 1 ​ ​ ?A. Log ⁡ 1 9 − Log ⁡ K \log \frac{1}{9} - \log K Lo G 9 1 ​ − Lo G K B. Log ⁡ 1 9 ÷ Log ⁡ K \log \frac{1}{9} \div \log K Lo G 9 1 ​ ÷ Lo G K C. Log ⁡ 1 9 ⋅ Log ⁡ K \log \frac{1}{9} \cdot \log K Lo G 9 1 ​ ⋅ Lo G K D. Log ⁡ 1 9 + Log ⁡ K \log \frac{1}{9} + \log K Lo G 9 1 ​ + Lo G K

Introduction

Logarithms are a fundamental concept in mathematics, and understanding their properties is crucial for solving various mathematical problems. In this article, we will explore the equivalent of the expression $\log \frac{\frac{1}{9}}{k}$ and examine the properties of logarithms that make this possible.

The Quotient Rule of Logarithms

The quotient rule of logarithms states that the logarithm of a quotient is equal to the logarithm of the dividend minus the logarithm of the divisor. Mathematically, this can be expressed as:

$$\log \frac{a}{b} = \log a - \log b$$

This rule is essential in simplifying complex logarithmic expressions and is a fundamental property of logarithms.

Applying the Quotient Rule

Now, let's apply the quotient rule to the given expression $\log \frac{\frac{1}{9}}{k}$. Using the quotient rule, we can rewrite this expression as:

$$\log \frac{\frac{1}{9}}{k} = \log \frac{1}{9} - \log k$$

This is a direct application of the quotient rule, where the dividend is $\frac{1}{9}$ and the divisor is $k$.

Evaluating the Options

Now that we have simplified the expression using the quotient rule, let's evaluate the options provided:

• Option A: $\log \frac{1}{9} - \log k$
• Option B: $\log \frac{1}{9} \div \log k$
• Option C: $\log \frac{1}{9} \cdot \log k$
• Option D: $\log \frac{1}{9} + \log k$

Based on our simplification using the quotient rule, we can see that Option A is the correct equivalent of the expression $\log \frac{\frac{1}{9}}{k}$.

Conclusion

In conclusion, the quotient rule of logarithms is a fundamental property that allows us to simplify complex logarithmic expressions. By applying this rule, we can rewrite the expression $\log \frac{\frac{1}{9}}{k}$ as $\log \frac{1}{9} - \log k$. This demonstrates the importance of understanding the properties of logarithms in solving mathematical problems.

Additional Tips and Tricks

• Recall the Product Rule: The product rule of logarithms states that the logarithm of a product is equal to the sum of the logarithms of the factors. Mathematically, this can be expressed as:

$$\log (ab) = \log a + \log b$$

This rule is essential in simplifying complex logarithmic expressions and is a fundamental property of logarithms.

• Recall the Power Rule: The power rule of logarithms states that the logarithm of a power is equal to the exponent multiplied by the logarithm of the base. Mathematically, this can be expressed as:

$$\log a^b = b \log a$$

This rule is essential in simplifying complex logarithmic expressions and is a fundamental property of logarithms.

Final Thoughts

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Q&A: Logarithmic Equivalents

Q: What is the quotient rule of logarithms?

A: The quotient rule of logarithms states that the logarithm of a quotient is equal to the logarithm of the dividend minus the logarithm of the divisor. Mathematically, this can be expressed as:

$$\log \frac{a}{b} = \log a - \log b$$

Q: How do I apply the quotient rule to simplify a logarithmic expression?

A: To apply the quotient rule, identify the dividend and the divisor in the logarithmic expression. Then, rewrite the expression using the quotient rule formula:

$$\log \frac{a}{b} = \log a - \log b$$

Q: What is the product rule of logarithms?

A: The product rule of logarithms states that the logarithm of a product is equal to the sum of the logarithms of the factors. Mathematically, this can be expressed as:

$$\log (ab) = \log a + \log b$$

Q: How do I apply the product rule to simplify a logarithmic expression?

A: To apply the product rule, identify the factors in the logarithmic expression. Then, rewrite the expression using the product rule formula:

$$\log (ab) = \log a + \log b$$

Q: What is the power rule of logarithms?

A: The power rule of logarithms states that the logarithm of a power is equal to the exponent multiplied by the logarithm of the base. Mathematically, this can be expressed as:

$$\log a^b = b \log a$$

Q: How do I apply the power rule to simplify a logarithmic expression?

A: To apply the power rule, identify the exponent and the base in the logarithmic expression. Then, rewrite the expression using the power rule formula:

$$\log a^b = b \log a$$

Q: What is the logarithmic equivalent of $\log \frac{\frac{1}{9}}{k}$?

A: Using the quotient rule, the logarithmic equivalent of $\log \frac{\frac{1}{9}}{k}$ is:

$$\log \frac{\frac{1}{9}}{k} = \log \frac{1}{9} - \log k$$

Q: How do I evaluate logarithmic expressions with multiple operations?

A: To evaluate logarithmic expressions with multiple operations, follow the order of operations (PEMDAS):

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: What are some common logarithmic identities?

A: Some common logarithmic identities include:

• $\log a^b = b \log a$
• $\log (ab) = \log a + \log b$
• $\log \frac{a}{b} = \log a - \log b$

Q: How do I use logarithmic identities to simplify expressions?

A: To use logarithmic identities to simplify expressions, identify the type of logarithmic operation being performed and apply the corresponding identity. For example, if you have a logarithmic expression with a quotient, you can use the quotient rule to simplify it.

Conclusion

In this Q&A article, we explored the properties of logarithms and how to apply them to simplify logarithmic expressions. By understanding the quotient rule, product rule, and power rule, you can simplify complex logarithmic expressions and evaluate them with ease.