Condense To A Single Logarithm: Ln ⁡ X + Ln ⁡ Y − Z \ln X + \ln Y - Z Ln X + Ln Y − Z A. Ln ⁡ X Y Z \ln \frac{xy}{z} Ln Z X Y ​ B. Ln ⁡ X Y Ln ⁡ Z \ln \frac{xy}{\ln Z} Ln L N Z X Y ​ C. Ln ⁡ ( X + Y − Z \ln (x + Y - Z Ln ( X + Y − Z ] D. ( Ln ⁡ X ) ( Ln ⁡ Y Z (\ln X)(\ln \frac{y}{z} ( Ln X ) ( Ln Z Y ​ ]

Introduction

Logarithmic expressions are a fundamental concept in mathematics, and condensing them into a single logarithm is an essential skill for students and professionals alike. In this article, we will explore the process of condensing logarithmic expressions, focusing on the given problem: $\ln x + \ln y - z$. We will examine the properties of logarithms and apply them to simplify the expression.

Properties of Logarithms

Before we dive into the problem, let's review the properties of logarithms that will be essential in condensing the expression:

• Product Property: $\log_a (xy) = \log_a x + \log_a y$
• Quotient Property: $\log_a \frac{x}{y} = \log_a x - \log_a y$
• Power Property: $\log_a x^n = n \log_a x$
• Addition Property: $\log_a x + \log_a y = \log_a (xy)$

Condensing the Expression

Now that we have reviewed the properties of logarithms, let's apply them to condense the expression $\ln x + \ln y - z$. We can start by using the Product Property to combine the first two terms:

$$\ln x + \ln y = \ln (xy)$$

Next, we can use the Quotient Property to rewrite the expression as a single logarithm:

$$\ln (xy) - z = \ln \frac{xy}{z}$$

However, we need to be careful when applying the properties of logarithms. The expression $-z$ is not a logarithm, so we cannot simply add it to the expression. Instead, we need to rewrite the expression as a single logarithm using the Power Property:

$$\ln \frac{xy}{z} = \ln \left(\frac{xy}{z}\right)^1 = \ln \frac{xy}{z}$$

Conclusion

In conclusion, the correct answer is $\ln \frac{xy}{z}$. We applied the properties of logarithms to condense the expression $\ln x + \ln y - z$ into a single logarithm. It's essential to carefully apply the properties of logarithms to avoid errors and ensure that the expression is simplified correctly.

Common Mistakes

When condensing logarithmic expressions, it's easy to make mistakes. Here are some common errors to watch out for:

• Incorrect application of properties: Make sure to carefully apply the properties of logarithms to avoid errors.
• Not simplifying the expression: Take the time to simplify the expression and ensure that it is in its simplest form.
• Not checking the answer: Double-check the answer to ensure that it is correct.

Practice Problems

To practice condensing logarithmic expressions, try the following problems:

• $\ln x + \ln y + z$
• $\ln \frac{x}{y} - \ln z$
• $\ln (xy) + \ln z$

Real-World Applications

Condensing logarithmic expressions has many real-world applications, including:

• Finance: Logarithmic expressions are used to calculate interest rates and investment returns.
• Science: Logarithmic expressions are used to model population growth and chemical reactions.
• Engineering: Logarithmic expressions are used to design and optimize systems.

Conclusion

In conclusion, condensing logarithmic expressions is an essential skill for students and professionals alike. By applying the properties of logarithms, we can simplify complex expressions and ensure that they are in their simplest form. Remember to carefully apply the properties of logarithms, simplify the expression, and check the answer to ensure that it is correct. With practice and patience, you will become proficient in condensing logarithmic expressions and be able to apply them to real-world problems.

Final Answer

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Introduction

Logarithmic expressions are a fundamental concept in mathematics, and condensing them into a single logarithm is an essential skill for students and professionals alike. In this article, we will explore the process of condensing logarithmic expressions, focusing on the properties of logarithms and providing a comprehensive Q&A guide.

Q&A: Condensing Logarithmic Expressions

Q: What is the product property of logarithms?

A: The product property of logarithms states that $\log_a (xy) = \log_a x + \log_a y$. This means that the logarithm of a product is equal to the sum of the logarithms of the individual factors.

Q: How do I apply the product property to condense a logarithmic expression?

A: To apply the product property, simply combine the logarithms of the individual factors using the addition property. For example, $\log_a (xy) = \log_a x + \log_a y$.

Q: What is the quotient property of logarithms?

A: The quotient property of logarithms states that $\log_a \frac{x}{y} = \log_a x - \log_a y$. This means that the logarithm of a quotient is equal to the difference of the logarithms of the individual factors.

Q: How do I apply the quotient property to condense a logarithmic expression?

A: To apply the quotient property, simply subtract the logarithm of the divisor from the logarithm of the dividend. For example, $\log_a \frac{x}{y} = \log_a x - \log_a y$.

Q: What is the power property of logarithms?

A: The power property of logarithms states that $\log_a x^n = n \log_a x$. This means that the logarithm of a power is equal to the exponent multiplied by the logarithm of the base.

Q: How do I apply the power property to condense a logarithmic expression?

A: To apply the power property, simply multiply the exponent by the logarithm of the base. For example, $\log_a x^3 = 3 \log_a x$.

Q: How do I condense a logarithmic expression with multiple terms?

A: To condense a logarithmic expression with multiple terms, apply the properties of logarithms in the following order:

1. Combine the logarithms of the individual factors using the product property.
2. Combine the logarithms of the individual factors using the quotient property.
3. Simplify the expression using the power property.

Q: What are some common mistakes to watch out for when condensing logarithmic expressions?

A: Some common mistakes to watch out for when condensing logarithmic expressions include:

• Incorrect application of properties
• Not simplifying the expression
• Not checking the answer

Q: How do I check my answer when condensing a logarithmic expression?

A: To check your answer, simply apply the properties of logarithms in reverse to ensure that the expression is simplified correctly.

Real-World Applications

Condensing logarithmic expressions has many real-world applications, including:

• Finance: Logarithmic expressions are used to calculate interest rates and investment returns.
• Science: Logarithmic expressions are used to model population growth and chemical reactions.
• Engineering: Logarithmic expressions are used to design and optimize systems.

Conclusion

In conclusion, condensing logarithmic expressions is an essential skill for students and professionals alike. By applying the properties of logarithms, we can simplify complex expressions and ensure that they are in their simplest form. Remember to carefully apply the properties of logarithms, simplify the expression, and check the answer to ensure that it is correct. With practice and patience, you will become proficient in condensing logarithmic expressions and be able to apply them to real-world problems.

Final Answer

The final answer is $\boxed{\ln \frac{xy}{z}}$.