What Is $3 \ln A + \ln B - \ln C$ Written As A Single Logarithm?A. Ln ⁡ 3 A B C \ln 3abc Ln 3 Ab C B. Ln ⁡ A 3 C B \ln \frac{a^3 C}{b} Ln B A 3 C ​ C. Ln ⁡ 3 A B C \ln \frac{3ab}{c} Ln C 3 Ab ​ D. Ln ⁡ A 3 B C \ln \frac{a^3 B}{c} Ln C A 3 B ​

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Introduction

Logarithmic expressions can be complex and difficult to work with, especially when they involve multiple terms. However, with the right techniques and formulas, we can simplify these expressions and rewrite them in a more manageable form. In this article, we will explore how to simplify the expression $3 \ln a + \ln b - \ln c$ and rewrite it as a single logarithm.

Understanding Logarithmic Properties

Before we dive into simplifying the expression, it's essential to understand the properties of logarithms. The most commonly used properties are:

  • Product Property: ln⁡(xy)=ln⁡x+ln⁡y\ln (xy) = \ln x + \ln y
  • Quotient Property: ln⁡(xy)=ln⁡x−ln⁡y\ln \left(\frac{x}{y}\right) = \ln x - \ln y
  • Power Property: ln⁡(xa)=aln⁡x\ln (x^a) = a \ln x

These properties will be crucial in simplifying the given expression.

Simplifying the Expression

Now that we have a good understanding of logarithmic properties, let's apply them to simplify the expression $3 \ln a + \ln b - \ln c$.

Using the Power Property, we can rewrite the first term as:

3ln⁡a=ln⁡a33 \ln a = \ln a^3

Now, the expression becomes:

ln⁡a3+ln⁡b−ln⁡c\ln a^3 + \ln b - \ln c

Using the Product Property, we can combine the first two terms:

ln⁡a3+ln⁡b=ln⁡(a3b)\ln a^3 + \ln b = \ln (a^3b)

Now, the expression becomes:

ln⁡(a3b)−ln⁡c\ln (a^3b) - \ln c

Using the Quotient Property, we can rewrite the expression as:

ln⁡(a3bc)\ln \left(\frac{a^3b}{c}\right)

Conclusion

In this article, we have successfully simplified the expression $3 \ln a + \ln b - \ln c$ and rewritten it as a single logarithm: ln⁡(a3bc)\ln \left(\frac{a^3b}{c}\right). This demonstrates the power of logarithmic properties in simplifying complex expressions.

Answer

The correct answer is:

  • D. ln⁡a3bc\ln \frac{a^3 b}{c}

This answer is based on the simplified expression we derived using logarithmic properties.

Final Thoughts

Introduction

In our previous article, we explored how to simplify the expression $3 \ln a + \ln b - \ln c$ and rewrite it as a single logarithm. In this article, we will provide a Q&A guide to help you better understand logarithmic expressions and how to simplify them.

Q&A

Q: What is the product property of logarithms?

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

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

A: To apply the product property, you need to identify the terms in the expression that can be multiplied together. Then, you can rewrite the expression using the product property formula: ln⁡(xy)=ln⁡x+ln⁡y\ln (xy) = \ln x + \ln y.

Q: What is the quotient property of logarithms?

A: The quotient property of logarithms states that ln⁡(xy)=ln⁡x−ln⁡y\ln \left(\frac{x}{y}\right) = \ln x - \ln y. This means that the logarithm of a quotient is equal to the difference of the logarithms of the individual terms.

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

A: To apply the quotient property, you need to identify the terms in the expression that can be divided together. Then, you can rewrite the expression using the quotient property formula: ln⁡(xy)=ln⁡x−ln⁡y\ln \left(\frac{x}{y}\right) = \ln x - \ln y.

Q: What is the power property of logarithms?

A: The power property of logarithms states that ln⁡(xa)=aln⁡x\ln (x^a) = a \ln 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 simplify a logarithmic expression?

A: To apply the power property, you need to identify the terms in the expression that can be raised to a power. Then, you can rewrite the expression using the power property formula: ln⁡(xa)=aln⁡x\ln (x^a) = a \ln x.

Q: Can I simplify a logarithmic expression with multiple terms?

A: Yes, you can simplify a logarithmic expression with multiple terms by applying the product, quotient, and power properties in the correct order.

Q: How do I determine the correct order of operations when simplifying a logarithmic expression?

A: To determine the correct order of operations, you need to 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 is the final answer to the expression $3 \ln a + \ln b - \ln c$?

A: The final answer to the expression $3 \ln a + \ln b - \ln c$ is ln⁡(a3bc)\ln \left(\frac{a^3b}{c}\right).

Conclusion

In this article, we have provided a Q&A guide to help you better understand logarithmic expressions and how to simplify them. By applying the product, quotient, and power properties, you can rewrite complex expressions in a more manageable form, making it easier to work with them.

Final Thoughts

Simplifying logarithmic expressions is an essential skill in mathematics, and it requires a good understanding of logarithmic properties. By following the steps outlined in this article, you can simplify even the most complex logarithmic expressions and become more confident in your math skills.