Which Expression Is Equivalent To Log ⁡ 3 C 9 \log _3 \frac{c}{9} Lo G 3 ​ 9 C ​ ?A. Log ⁡ 3 C + Log ⁡ 3 ( 9 \log _3 C + \log _3(9 Lo G 3 ​ C + Lo G 3 ​ ( 9 ] B. Log ⁡ 3 ( 9 ) + Log ⁡ 3 ( C \log _3(9) + \log _3(c Lo G 3 ​ ( 9 ) + Lo G 3 ​ ( C ] C. Log ⁡ 3 C − Log ⁡ 3 ( 9 \log _3 C - \log _3(9 Lo G 3 ​ C − Lo G 3 ​ ( 9 ] D. Log ⁡ 3 ( 9 ) − Log ⁡ 3 ( C \log _3(9) - \log _3(c Lo G 3 ​ ( 9 ) − Lo G 3 ​ ( C ]

In mathematics, logarithms are a fundamental concept used to solve equations and manipulate expressions. A logarithm is the inverse operation of exponentiation, and it helps us find the power to which a base number must be raised to obtain a given value. In this article, we will explore the concept of logarithmic expressions and determine which expression is equivalent to $\log _3 \frac{c}{9}$.

Logarithmic Properties

Before we dive into the problem, let's review some essential logarithmic properties:

• Product Rule: $\log_b (xy) = \log_b x + \log_b y$
• Quotient Rule: $\log_b \frac{x}{y} = \log_b x - \log_b y$
• Power Rule: $\log_b x^y = y \log_b x$

These properties will help us simplify the given expression and determine its equivalent form.

Simplifying the Expression

The given expression is $\log _3 \frac{c}{9}$. To simplify this expression, we can use the Quotient Rule, which states that $\log_b \frac{x}{y} = \log_b x - \log_b y$. Applying this rule, we get:

$$\log _3 \frac{c}{9} = \log _3 c - \log _3 9$$

Now, let's focus on the second term, $\log _3 9$. We can use the Power Rule, which states that $\log_b x^y = y \log_b x$. Since $9 = 3^2$, we can rewrite the expression as:

$$\log _3 9 = \log _3 (3^2) = 2 \log _3 3$$

However, $\log _3 3 = 1$, so we can simplify the expression further:

$$\log _3 9 = 2 \log _3 3 = 2 \cdot 1 = 2$$

Now, let's substitute this value back into the original expression:

$$\log _3 \frac{c}{9} = \log _3 c - \log _3 9 = \log _3 c - 2$$

Comparing with the Options

Now that we have simplified the expression, let's compare it with the given options:

A. $\log _3 c + \log _3(9)$ B. $\log _3(9) + \log _3(c)$ C. $\log _3 c - \log _3(9)$ D. $\log _3(9) - \log _3(c)$

Based on our simplification, we can see that option C is the correct answer: $\log _3 c - \log _3(9)$.

Conclusion

In this article, we explored the concept of logarithmic expressions and simplified the expression $\log _3 \frac{c}{9}$. We used the Quotient Rule and the Power Rule to simplify the expression and determined that the equivalent form is $\log _3 c - \log _3 9$. We then compared this expression with the given options and concluded that option C is the correct answer.

Final Answer

The final answer is $\boxed{C}$.

Additional Resources

For more information on logarithmic expressions and properties, check out the following resources:

• Khan Academy: Logarithms
• Mathway: Logarithmic Properties
• Wolfram MathWorld: Logarithm

In the previous article, we explored the concept of logarithmic expressions and simplified the expression $\log _3 \frac{c}{9}$. In this article, we will answer some frequently asked questions about logarithmic expressions and provide additional insights into this fundamental concept in mathematics.

Q: What is the difference between a logarithm and an exponent?

A: A logarithm is the inverse operation of an exponent. While an exponent tells us how many times a base number must be multiplied by itself to obtain a given value, a logarithm tells us the power to which a base number must be raised to obtain a given value.

Q: What are the basic logarithmic properties?

A: The basic logarithmic properties are:

• Product Rule: $\log_b (xy) = \log_b x + \log_b y$
• Quotient Rule: $\log_b \frac{x}{y} = \log_b x - \log_b y$
• Power Rule: $\log_b x^y = y \log_b x$

These properties help us simplify logarithmic expressions and solve equations involving logarithms.

Q: How do I simplify a logarithmic expression?

A: To simplify a logarithmic expression, you can use the following steps:

1. Identify the base and the argument of the logarithm.
2. Use the Quotient Rule to simplify the expression if it involves a fraction.
3. Use the Power Rule to simplify the expression if it involves an exponent.
4. Use the Product Rule to simplify the expression if it involves a product.

Q: What is the logarithm of a negative number?

A: The logarithm of a negative number is undefined in the real number system. However, in the complex number system, the logarithm of a negative number can be defined using the formula $\log_b (-x) = \log_b x + i \pi$.

Q: Can I use a calculator to evaluate a logarithmic expression?

A: Yes, you can use a calculator to evaluate a logarithmic expression. Most calculators have a built-in logarithm function that allows you to enter the base and the argument of the logarithm and obtain the result.

Q: How do I graph a logarithmic function?

A: To graph a logarithmic function, you can use the following steps:

1. Identify the base and the argument of the logarithm.
2. Determine the domain and range of the function.
3. Plot the points on the graph using the formula $y = \log_b x$.
4. Draw a smooth curve through the points to obtain the graph of the function.

Q: What are some common applications of logarithmic expressions?

A: Logarithmic expressions have many applications in various fields, including:

• Finance: Logarithmic expressions are used to calculate interest rates, investment returns, and stock prices.
• Science: Logarithmic expressions are used to calculate pH levels, sound levels, and light intensities.
• Engineering: Logarithmic expressions are used to calculate stress, strain, and pressure in materials.

Conclusion

In this article, we answered some frequently asked questions aboutmic expressions and provided additional insights into this fundamental concept in mathematics. We hope that this article has helped you deepen your understanding of logarithmic expressions and their applications in various fields.

Final Answer

The final answer is $\boxed{C}$.

Additional Resources

For more information on logarithmic expressions and properties, check out the following resources:

• Khan Academy: Logarithms
• Mathway: Logarithmic Properties
• Wolfram MathWorld: Logarithm

These resources provide a comprehensive overview of logarithmic expressions and properties, and can help you deepen your understanding of this fundamental concept in mathematics.